Quotienting conjecture for Picard groups of double covers of punctured curves

Let $C$ be a smooth projective curve over $\mathbb F_q$, let $S\subset C$ be a reduced effective divisor, and consider smooth projective double covers $D\to C$ unramified over $S$. Put $T=\pi^{-1}(S)$ and $u=\#T-\#S$. Let $\mu^g_{C,S}$ be the counting measure of the enhanced relative Picard groups $(\operatorname{Pic}(D/C)_\ell,\omega_{D/C},\psi_{D/C})$ as the genus-$g$ covers vary, and let $Q^u\mu$ denote the measure obtained by quotienting by $u$ random elements. Punctured-curve quotienting conjecture. As $g\to\infty$, the measures $\mu^g_{C,S}$ converge to $Q^u\mu$ in the weak-* topology. This models the contribution of punctures by random quotienting and motivates the exponent $t=[K:\mathbb Q]/2$ in the number-field conjecture.