Equality conjecture for tensor ideals in tilting modules

Let $G$ be an algebraic group and $p$ a prime. Let $\mathsf{Tilt} G$ be the category of tilting $G$-modules; let $I_n(G)$ and $\overline I_n(G)$ be the indicated ideals of objects, and let $\mathcal{I}_n(G)$ and $\overline{\mathcal{I}}_n(G)$ be the corresponding ideals of morphisms. Let $\Phi^+$ be the set of positive roots, and let $N_k$ and $\mathcal{N}_k$ be the thick tensor ideal of objects and tensor ideal of morphisms indexed by $k$.

Tensor-ideal equality conjecture. For any $G$ and $p$, all of the following hold:

1. $I_n(G)=\overline I_n(G)\cap\operatorname{Ob}(\mathsf{Tilt} G)$, and consequently $\mathcal{I}_n(G)=\overline{\mathcal{I}}_n(G)\cap\operatorname{Mor}(\mathsf{Tilt} G)$;
2. $I_n(G)=N_{(n-1)|\Phi^+|+1}$ in $\mathsf{Tilt} G$;
3. $\mathcal{I}_n(G)=\mathcal{N}_{(n-1)|\Phi^+|+1}$ in $\mathsf{Tilt} G$.

The paper motivates these equalities by noting that the corresponding restrictions to $\mathcal{T}_n$ are already known from the cited results. The source presents the equalities as a conjecture and gives no resolution status for them in the supplied text.