The irreducible-component conjecture for symmetric Seesaw representations

Let $\mathcal{A}$ be the Seesaw algebra, let $V$ be the representation space, and let $R(\mathcal{A},V)^{\langle-,-\rangle,\varepsilon}$ be the symmetric representation variety. For representations $M$ and $N$, write $M\leq_{\deg}N$ for ordinary degeneration and $M\leq_{\deg}^{\varepsilon}N$ for symmetric degeneration.

Irreducible-component conjecture. If $I$ is an irreducible component of $R(\mathcal{A},V)^{\langle-,-\rangle,\varepsilon}$ and $M,N\in I$, then

$$M\leq_{\deg}^{\varepsilon}N\quad\Longleftrightarrow\quad M\leq_{\deg}N.$$

The conjecture proposes that Main Question 2 has a positive answer on every irreducible component of the symmetric representation variety. The paper establishes that the question is not true in general; symmetric representation varieties are irreducible in type $C$, while the situation in type $B$ is uncertain, and type $D$ has two dense orbits.