Euclidean distance degree for diagonal-zero rank-two varieties

Let $\mathcal{L}_2^S\subset\mathbb{C}^{m\times n}$ be the variety of matrices of rank at most $2$ with diagonal zero pattern $S=\{(1,1),\ldots,(s,s)\}$, where $s\in[4]$, and set $l=\min(m,n)$. Diagonal-zero ED-degree conjecture.

$$\operatorname{EDdegree}(\mathcal{L}_2^S)= \begin{cases} 3(l-1)^2-2(l-1) & \text{if }s=1,\\ 18(l-2)^2+6(l-2)+1 & \text{if }s=2\text{ and }m\ne n,\\ 18(l-2)^2+10(l-2)+1 & \text{if }s=2\text{ and }m=n,\\ 108(l-3)^2+144(l-3)+30 & \text{if }s=3\text{ and }m=n,\\ 648(l-4)^2+1600(l-4)+488 & \text{if }s=4\text{ and }m=n. \end{cases}$$

These formulas are experimentally supported for the cases tabulated in the paper. The supplied text does not establish them in general, and the displayed cases do not specify formulas for every combination of $m,n,s$.