Almost maximally rigid non-closedness conjecture

Let $\operatorname{LS}_n(r,s)$ denote the set of matrices that can be written as the sum of an $n\times n$ matrix of rank at most $r$ and an $n\times n$ matrix with at most $s$ nonzero entries. Almost maximally rigid non-closedness conjecture. The low-rank plus sparse set $\operatorname{LS}_n(r,s)$ is not closed provided

$$n \geq r + (s+1)^{1/2},$$

for $s \in [1,(n-1)^2-1]$ and $r \in [1,n-2]$. This would generalize the established non-closedness of $\operatorname{LS}_3(1,3)$ to the highest sparsities allowed by the stated bounds, but the general case remains open.