Smoothness conjecture for Fano schemes of bounded-rank symmetric matrices

Let $\mathrm{\textbf{F}}_{k}(\mathrm{SD}_n^r)$ denote the Fano scheme of $k$-planes contained in the space of symmetric $n\times n$ matrices of rank at most $r$. Let $\kappa(s)$ be the threshold defined in the paper for nested $s$-compression spaces. Smoothness conjecture. The scheme $\mathrm{\textbf{F}}_{k}(\mathrm{SD}_n^r)$ is smooth if and only if $r$ is odd and

$$\max\left\{ \kappa(0),\kappa\left(\frac{r-3}{2}\right) \right\} < k \leq \kappa\left(\frac{r-1}{2}\right).$$

The conjecture aims to complete the partial smoothness results established for these Fano schemes by combining them with the paper's connectedness results; the general characterization remains open.