Conjecture on irreducible components of relative tangency varieties

Let $X_r$ be the determinantal variety of matrices of rank at most $r$, let $S$ be the space of symmetric $(N+1)\times(N+1)$ matrices, and let $S_r=X_r\cap S$. For $r\in[N+1]$, define

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

With the same notation for the corresponding projective varieties, write $A=\bigwedge^2\mathbb{C}^{N+1}$.

Conjecture on the components of $\mathcal{SC}_r$. For all $r\in[N+1]$:

1. $\mathcal{SC}_1=S=S^2\mathbb{C}^{N+1}$ is an irreducible component of $\mathcal{SC}_r$ for every $r\in[N+1]$.
2. $(X_{N-r})_{S_{N-r}}^\vee$ is an irreducible component of $\mathcal{SC}_r$ for every $2\le r\le N+1$.
3. $A=\bigwedge^2\mathbb{C}^{N+1}$ is an irreducible component of $\mathcal{SC}_r$ for all $N\ge3$ and even $r\in[N]$.

These component claims are suggested by symbolic computations for small values of $N$, where additional irreducible components can occur and the varieties $\mathcal{SC}_i$ need not be nested. The conjecture concerns the persistent components arising from the symmetric and skew-symmetric matrix loci and the relative dual determinantal varieties.