Genericity conjecture for the favorable spectral-data locus

Let $X$ be an algebraic surface, and let $\mathscr B_X^{\Diamond}(k)$ be the distinguished subset of the postulated Hitchin base $\mathscr B_X(k)$ defined in the surrounding discussion. Genericity conjecture. The subset $\mathscr B_X^{\Diamond}(k)$ is a non-empty open subset of $\mathscr B_X(k)$. The source says this is expected for most algebraic surfaces and relates its non-emptiness to poorly understood questions about zero loci of symmetric differentials.