Largest collection of square matrices with zero on diagonal of products

Question

Let $n$ be a positive integer. What is the largest integer $k$ for which there exist $n\times n$ matrices $M_1,M_2,\ldots,M_k$ and $N_1,N_2,\ldots,N_k$ such that for all $i$ and $j$, the matrix product $M_iN_j$ has a zero entry somewhere on its diagonal if and only if $i\neq j$.