Quadratic ED-degree growth for square diagonal-zero rank-two varieties

Let $\mathcal{L}_2^S\subset\mathbb{C}^{m\times m}$ be the variety of square matrices of rank at most $2$ with zero pattern $S=\{(1,1),\ldots,(s,s)\}$ for some $s\ge2$. Quadratic-growth conjecture. For some constant $c$,

$$\operatorname{EDdegree}(\mathcal{L}_2^S)=3^s2^{s-1}(n-s)^2+s^{s-1}(s+1)^{\lceil s/2\rceil}(n-s)+c.$$

The conjecture is an extrapolation from the experimental tables and asserts quadratic dependence on the matrix size, with the displayed leading and linear coefficients. The constant $c$ is not specified, and no proof or resolution is supplied.