Two-part criterion for odd-characteristic subgroup rigidity

Let $F$ be a free group and let $p$ be an odd prime. Let $H$ be a finitely generated subgroup of $F$, and let $\widetilde H$ be its $L^2$-closure. The two-part odd-characteristic criterion.

$$\beta_0^{\mathbb{F}_p[F]}(I_{\mathbb{F}_p[F]}/{}^FI_{\mathbb{F}_p[H]})=\beta_0^{\mathbb{Q}[F]}(I_{\mathbb{Q}[F]}/{}^FI_{\mathbb{Q}[H]}).$$

2. The left $\mathbb{F}_p[F]$-module $I_{\mathbb{F}_p[F]}/{}^FI_{\mathbb{F}_p[\widetilde H]}$ is $\mathcal{D}_{\mathbb{F}_p[F]}$-torsion-free.

These are presented as the two subproblems underlying odd-characteristic subgroup rigidity of free groups. They remain open in the supplied text.