Conjecture on irreducible components of varieties with symmetric compound matrices

Let $N$ be the matrix-size parameter, let $[N+1]=\{1,\ldots,N+1\}$, and let $S=S^2\mathbb{C}^{N+1}$ be the subspace of symmetric $(N+1)\times(N+1)$ matrices. For $r\in[N+1]$, define

$$\mathcal{SC}_r=\{B\in\mathbb{C}^{(N+1)\times(N+1)}\mid C_r(B)\text{ is symmetric}\},$$

where $C_r(B)$ is the $r$th compound matrix of $B$, and write $(X_j)_{S_j}^\vee$ for the relative dual variety introduced earlier. Let $A=\bigwedge^2\mathbb{C}^{N+1}$ denote the space of skew-symmetric matrices. Irreducible-component conjecture. For every $r\in[N+1]$: (i) $\mathcal{SC}_1=S=S^2\mathbb{C}^{N+1}$ is an irreducible component of $\mathcal{SC}_r$; (ii) $(X_{n-r})_{S_{n-r}}^\vee$ is an irreducible component of $\mathcal{SC}_r$ for every $2\le r\le N+1$; and (iii) $A=\bigwedge^2\mathbb{C}^{N+1}$ is an irreducible component of $\mathcal{SC}_r$ for all $N\ge 3$ and even $r\in[N]$. The conjecture is motivated by symbolic computations for $N=2$ and $N=3$, which exhibit several of these components, while the preceding example also shows that the varieties $\mathcal{SC}_i$ need not be nested. Determining all irreducible components in general remains open.