Symplectic Pfaffian conjecture for spherical dual representations

Let $X=G/H$ be an affine homogeneous spherical variety with abelian regular centralizers and no type $N$ roots. Let $S_X$ be the corresponding symplectic representation of $G_X^\vee$, let $d\rho:\mathfrak g_X^\vee\to\mathfrak{sp}(S_X)$ be the induced Lie-algebra map, and let $\det$ be the determinant function on $\mathfrak{sp}(S_X)$. Symplectic Pfaffian conjecture. There exists a function $\mathrm{Pf}_X$ on $\mathfrak g_X^\vee$, unique up to sign, such that

$$d\rho^*(\det)=(\mathrm{Pf}_X)^2.$$

The paper verifies this in the polarized case and in examples, while the general assertion is presented as a conjecture.