Let $A$ be a finite dimensional algebra. The big finitistic dimension of $A$ is

$$\operatorname{FinDim}(A)=\sup\{\operatorname{pd}(M)\mid M\in \text{Mod-}A \text{ and } \operatorname{pd}(M)<\infty\},$$ where $\operatorname{pd}(M)$ is the projective dimension of $M$, $\text{Mod-}A$ is the category of right $A$-modules.

Problem: Does there exist a finite dimensional algebra $A$ such that $\operatorname{FinDim}(A)=\infty?$

http://cn.arxiv.org/abs/1804.09801 (see Definition 4.1.)