Degree and sign independence conjecture for automorphic \a3-invariants

Let $G$ be a reductive group, let $\a3$ be a sign character, let $d$ be an integer with $0\leq d\leq\delta$, and let $i\in\Delta$ be a root. The automorphic invariant $\mathcal{L}_{i}^{(d)}(\pi,\mathfrak{p})^{\epsilon}$ is a subspace of $\operatorname{Hom}_{\operatorname{ct}}(F_{\mathfrak{p}}^{\times},\Omega)$. Degree and sign independence conjecture. For every root $i\in\Delta$, $\mathcal{L}_{i}^{(d)}(\pi,\mathfrak{p})^{\epsilon}$ has codimension one for all $0\leq d\leq\delta$ and every sign character $\epsilon$; it is independent of $d$ and of $\epsilon$. The preceding proposition proves only codimension at least one in general, with exact codimension one in degrees $0$ and $\delta$ when $m_\pi=1$; the asserted independence remains conjectural.