For every large enough $m\in\Bbb N$ are there $c=\alpha m$ (for some fixed $\alpha>0$) square matrices $A_1,\dots,A_c$ that commute with each other with nonzero product ($\forall i,j\in\{1,\dots,t\}$ distinct, $A_iA_j=A_jA_i\neq 0$ holds) and satisfy $\forall i\in\{1,\dots,c\}$ $A_i^2=0$? What is smallest size of matrices (can they be $\beta m\times\beta m$ for some fixed $\beta>0$)?

I am ok with coefficients in any field with characteristic 0 or $O(2^{m^\gamma})$ for a fixed $\gamma>0$.

Similar problem: https://math.stackexchange.com/questions/1710735/commutating-nilpotent-operators.