Maximum-size conjecture for strongly identity-forcing matrices

Let $I_k$ be the $k\times k$ identity matrix. For an $n\times n$ matrix, let $\mathrm{M}(n,I_k)$ denote the maximum possible number of $1$-entries in a strongly $I_k$-forcing matrix. Maximum-size conjecture for strongly identity-forcing matrices. If $n\geq k\geq 3$, then

$$\mathrm{M}(n,I_k)=n^2-(2k-3)n-(2k-k^2).$$

The formula is motivated by an explicit strongly $I_k$-forcing construction and is known for $k=2$ and $k=3$; its validity for all $n\geq k\geq 3$ remains open.