Unless stated otherwise, in these cases the conjecture is know to be induced by a `split endomorphism two-sided tilting complex'.

Thanks to Naoko Kunugi for help in preparing this page.

- All blocks with cyclic defect group.
- All blocks of
*p*-solvable groups. - All blocks with defect group C
_{2}×C_{2}. - All principal blocks with defect group C
_{3}×C_{3}.- Koshitani and Kunugi [pre-KosKun1]. This depends on the classification of finite simple groups, and uses results of Okuyama and others mentioned below for specific groups, and general methods due to Marcus [Mar8].

- All blocks of symmetric groups with defect groups of order
*p*^{2}.- Chuang [Chu2].

- All blocks of symmetric groups with abelian defect groups of order at
most
*p*^{5}.- Chuang and Kessar [pre-ChuKes].

- The principal block of A
_{4},*p*=2. (Defect group C_{2}×C_{2}.)- Rickard [Ric6].

- The principal blocks of A
_{6}, A_{7}and A_{8},*p*=3. (Defect group C_{3}×C_{3}.)- Okuyama [pre-Oku1].

- The principal block of the Mathieu group M
_{11},*p*=3. (Defect group C_{3}×C_{3}.)- Okuyama [pre-Oku1].

- The principal block of the Mathieu group M
_{22},*p*=3. (Defect group C_{3}×C_{3}.)- Okuyama [pre-Oku1].

- The principal block of the Mathieu group M
_{23},*p*=3. (Defect group C_{3}×C_{3}.)- Okuyama [pre-Oku1].

- The principal block of the Higman-Sims group HS,
*p*=3. (Defect group C_{3}×C_{3}.)- Okuyama [pre-Oku1].

- The principal block of the Janko group J
_{1},*p*=2. (Defect group C_{2}×C_{2}×C_{2}.)- Gollan and Okuyama [pre-GolOku].

- The principal block of the Hall-Janko group HJ=J
_{2},*p*=5. (Defect group C_{5}×C_{5}.)- Holloway [pre-Hol].

- The non-principal block of the O'Nan group O'N with defect group
C
_{3}×C_{3}.- Koshitani, Kunugi and Waki [pre-KosKunWak].

- The non-principal block of the Higman-Sims group HS with defect group
C
_{3}×C_{3}.- Holm [Hol1], Koshitani, Kunugi and Waki [pre-KosKunWak].

- The non-principal block of the double cover 2.J
_{2}of the Hall-Janko group with defect group C_{5}×C_{5}.- Holloway [pre-Hol].

- The principal block of PSL
_{2}(*p*^{n}) in characteristic*p*. (Defect group C_{p}^{n}.)- Okuyama [pre-Oku2] for the general case; earlier Chuang [Chu1] for the case n=2.

- The non-principal block with full defect of
SL
_{2}(*p*^{2}) in characteristic*p*. (Defect group C_{p}×C_{p}.)- Holloway [pre-Hol].

- The principal block in characteristic
*p*of the group G=**G**^{F}of rational points of a connected reductive group**G**defined over the field**F**_{q}of*q*elements, where*p*is a prime dividing*q*-1 but not dividing the order of the Weyl group of G.- This follows from a result of Puig
[Pui4]
([Remark]). In this case the
blocks
*A*and*B*are Morita equivalent.This general theorem includes, for

*p*=3, some cases where G has Sylow 3-subgroup C_{3}×C_{3}, which were needed for the theorem of Koshitani and Kunugi. Namely: G=Sp_{4}(q), where*q*=4 or 7 (mod 9); G=PSU_{4}(q^{2}) or PSU_{5}(q^{2}), where*q*=4 or 7 (mod 9) [see also Koshitani and Miyachi [pre-KosMiy]]. Of course, Puig's theorem also deals with the case*q*=1 (mod 9), but then the Sylow*p*-subgroup is larger than C_{3}×C_{3}.

- This follows from a result of Puig
[Pui4]
([Remark]). In this case the
blocks
- The principal blocks of the Ree groups
R(3
^{2n+1})=^{2}G_{2}(3^{2n+1}),*p*=2. (Defect group C_{2}×C_{2}×C_{2}.)- It follows from work of Landrock and Michler [LanMic]
that the principal blocks of R(3
^{2n+1}) are all Morita equivalent (for different values of*n*). This reduces the conjecture to the case*n*=0, which is known, since R(3)=SL_{2}(8).3.

- It follows from work of Landrock and Michler [LanMic]
that the principal blocks of R(3
- The principal blocks of SU
_{3}(q^{2}), where*q*is a prime power, and*p*>3 is a prime dividing*q*+1. (Defect group C_{r}×C_{r}, where*r*=*p*^{a}is the largest power of*p*dividing*q*+1.)- Kunugi and Waki [KunWak].

- The principal blocks of PSL
_{3}(q),*p*=3, where*q*=4 or 7 (mod 9). (Defect group C_{3}×C_{3}.)- Kunugi [Kun3] proves that
these blocks are all Morita equivalent. The conjecture then follows
since Okuyama [pre-Oku1] proved the
case of PSL
_{3}(4).

- Kunugi [Kun3] proves that
these blocks are all Morita equivalent. The conjecture then follows
since Okuyama [pre-Oku1] proved the
case of PSL
- The principal blocks of Sp
_{4}(q),*p*=3, where*q*=2 or 5 (mod 9). (Defect group C_{3}×C_{3}.)- It follows from work of Okuyama and Waki [OkuWak]
that these blocks are all Morita
equivalent. The conjecture then follows, because Sp
_{4}(2) is isomorphic to the symmetric group S_{6}.

- It follows from work of Okuyama and Waki [OkuWak]
that these blocks are all Morita
equivalent. The conjecture then follows, because Sp
- The principal block of Sp
_{4}(4),*p*=5. (Defect group C_{5}×C_{5}.)- Holloway [pre-Hol].

- The principal blocks of PSU
_{3}(q^{2}),*p*=3, where*q*=2 or 5 (mod 9). (Defect group C_{3}×C_{3}.)- Koshitani and Kunugi [pre-KosKun2] prove that the
blocks
*A*and*B*are Morita equivalent.

- Koshitani and Kunugi [pre-KosKun2] prove that the
blocks
- The principal blocks of GL
_{4}(q),*p*=3, where*q*=2 or 5 (mod 9). (Defect group C_{3}×C_{3}.)- Koshitani and Miyachi [KosMiy]
show that these blocks are all Morita equivalent.
The conjecture then follows
since Okuyama [pre-Oku1]
proved the
case of GL
_{4}(2), which is isomorphic to A_{8}.

- Koshitani and Miyachi [KosMiy]
show that these blocks are all Morita equivalent.
The conjecture then follows
since Okuyama [pre-Oku1]
proved the
case of GL
- The principal blocks of GL
_{5}(q),*p*=3, where*q*=2 or 5 (mod 9). (Defect group C_{3}×C_{3}.)- Koshitani and Miyachi [KosMiy]
prove that the
blocks
*A*and*B*are Morita equivalent.

- Koshitani and Miyachi [KosMiy]
prove that the
blocks
- Unipotent blocks of weight 2 of general linear groups in non-defining
characteristic p.
(Defect group C
_{r}×C_{r}, for*r*=*p*^{a}some power of*p*.)- Hida and Miyachi [pre-HidMiy], Turner [pre-Tur].