Chi-independence conjecture for BPS sheaves in families of Higgs bundles

Let $f\colon\mathcal{C}\to S$ be a family of smooth projective curves of genus $g$ over a smooth connected base $S$. Let $\mathfrak{M}_{r,d}^{\operatorname{Dol}}(\mathcal{C})$ be the moduli stack of semistable rank-$r$, degree-$d$ Higgs bundles, let $\mathcal{M}_{r,d}^{\operatorname{Dol}}(\mathcal{C})$ be its good moduli space, and let $B_S$ be the Hitchin base, with

$$\mathfrak{M}_{r,d}^{\operatorname{Dol}}(\mathcal{C})\xrightarrow{\mathtt{JH}}\mathcal{M}_{r,d}^{\operatorname{Dol}}(\mathcal{C})\xrightarrow{h}B_S\xrightarrow{\operatorname{pr}}S.$$

Chi-independence conjecture. There exists a BPS sheaf in families $\mathcal{BPS}_{r,d}(\mathcal{C})\in\operatorname{MHM}(\mathcal{M}_{r,d}^{\operatorname{Dol}}(\mathcal{C}))[\dim S]$ such that for every $s\in S$, the exceptional restriction to the fiber satisfies

$$\imath_{\mathcal{M},s}^!\mathcal{BPS}_{r,d}^{\operatorname{Dol}}(\mathcal{C})\cong\mathcal{BPS}_{r,d}^{\operatorname{Dol}}(\mathcal{C}_s).$$

Moreover, $h_*\mathcal{BPS}_{r,d}^{\operatorname{Dol}}(\mathcal{C})$ has locally constant cohomology sheaves and is independent of $d$. This conjecture extends chi-independence from individual curves to families.