The non-vanishing conjecture at twice the rank for product actions

Let $G$ be a non-compact simple group over a local field $F$ with finite center and rank $r$. The twice-the-rank non-vanishing conjecture. For every sufficiently large $p$,

$$H_c^{2r}(G\times G,L^p(G))\ne0.$$

A similar conclusion is conjectured for all lattices in $G$. This complements the established vanishing below degree $2r$ for the product action and predicts the first non-zero degree at the doubled rank; the source gives no resolution status.