Graded-universality conjecture for Hall categories of nongeneric curves

Let $k=\mathbb{C}$, let $X$ be a smooth projective curve of genus $g$, let $T^a_g$ be the torus associated with the Frobenius-eigenvalue parameter space, and let $\boldsymbol{\chi}(T^a_g)$ be its character group. Let $\mathcal{Q}_X$ be the Hall category and let $\mathcal{K}_X$ be the associated graded Grothendieck group; let $\mathcal{K}_g$ denote the universal genus-$g$ object. Graded-universality conjecture. There exists a natural grading by $\boldsymbol{\chi}(T^a_g)$ on $\mathcal{Q}_X$, and the associated graded Grothendieck group satisfies

$$\mathcal{K}_X\simeq\mathcal{K}_g.$$

This proposal is meant to extend the generic-curve universality picture to nongeneric curves. The source presents it as a further expectation and gives no proof or status beyond that.