Euclidean distance degree for symmetric diagonal-zero rank-two varieties

Let $\mathcal{L}_2^S\subset\operatorname{Sym}_n(\mathbb{C})$ be the variety of symmetric matrices of rank at most $2$ with diagonal zero pattern $S=\{(1,1),\ldots,(s,s)\}$, where $s\in[4]$. Symmetric diagonal-zero ED-degree conjecture.

$$\operatorname{EDdegree}(\mathcal{L}_2^S)= \begin{cases} 3(n-1)-2 & \text{if }s=1,\\ 9(n-2)-2 & \text{if }s=2,\\ 27(n-3)+4 & \text{if }s=3,\\ 81(n-4)+28 & \text{if }s=4. \end{cases}$$

The formulas agree with the reported computations for the symmetric examples in the table. Their validity beyond those computations is not established in the supplied text.