Universality conjecture for generic-curve Hall algebras

Let $X$ and $X'$ be generic smooth projective curves of the same genus $g$, and let $\mathcal{K}_X$ and $\mathcal{K}_{X'}$ be their graded Grothendieck algebras over $\operatorname{Rep}(T^g_a)$. Generic-curve universality conjecture. For any two generic curves $X$ and $X'$,

$$\mathcal{K}_X\simeq\mathcal{K}_{X'}.$$

This asserts that the formal Hall algebra depends only on the genus among generic curves. The source says the claim is known for genus zero and one, but leaves higher-genus cases open.