Bobiński–Malicki conjecture on derived equivalences among the algebras A(p) and B(p,r)

For $p \in \mathbb{Z}_{\geq 0}$, $p' \in \mathbb{Z}_{\geq 1}$, and $r \in [0,p]$, let $A(p)$ and $B(p,r)$ be the algebras defined in the paper, and let ${\mathcal D}^b(-)$ denote their bounded derived categories of finite-dimensional modules.

Bobiński–Malicki conjecture.

(a) For all $p \in \mathbb{Z}_{\geq 0}$, $p' \in \mathbb{Z}_{\geq 1}$, and $r \in [0,p]$,

$${\mathcal D}^b(B(p,r)) \ncong {\mathcal D}^b(A(p')).$$

(b) If

$${\mathcal D}^b(B(p,r)) \cong {\mathcal D}^b(B(p',r')),$$

then $p=p'$ and $r=r'$.

The conjecture asserts that distinct algebras in the families $A(p)$ and $B(p,r)$ are not derived equivalent. It was proved by Bobiński, building on earlier work of Amiot.