The Picard-generation conjecture for Hurwitz spaces

Work over an algebraically closed field of characteristic zero. Let $\widetilde H_{d,g}$ be the Hurwitz space of degree-$d$ covers of smooth genus-$0$ curves by irreducible genus-$g$ curves, with unparametrized source and target. Let $\kappa$ and $\xi$ denote the tautological divisor classes on $\widetilde H_{d,g}$, and let $\Delta$ be the divisor parametrizing covers whose source curve is singular. Here $\operatorname{Pic}_{\mathbf Q}(\widetilde H_{d,g})$ denotes the rational Picard group. Picard-generation conjecture. The divisor classes $\kappa$, $\xi$, and $\Delta$ generate

$$\operatorname{Pic}_{\mathbf Q}(\widetilde H_{d,g}).$$

This predicts that these natural tautological and boundary classes exhaust the rational Picard group of the Hurwitz space, at least modulo torsion. The source provides no resolution status for the conjecture.