The Picard rank conjecture for simply branched Hurwitz spaces

Let $d>0$ and let $n$ be even with $n\geq 2d-2$. Let $c\subset S_d$ be the conjugacy class of transpositions, and let $[[\operatorname{CHur}^{S_d,c}_{\mathbb P^1,n}/S_d]/\operatorname{PGL}_2]$ denote the Hurwitz stack of geometrically connected degree-$d$ covers of genus-$0$ curves with branch-point inertia in $c$. Write $\pic(-)$ for its Picard group. Picard rank conjecture. For all such $d$, $n$, and $c$,

$$\pic([ [\operatorname{CHur}^{S_d,c}_{\mathbb P^1,n}/S_d]/\operatorname{PGL}_2])\otimes\mathbb Q\simeq 0.$$

This is the original Picard rank conjecture for simply branched covers, where $n=2g-2+2d$ and $g\geq0$. Mullane proved it whenever $n\leq4d-2$; the general case remains open.