Base-change compatibility conjecture for automorphic \a3-invariants

Let $E/F$ be a solvable extension, let $\mathfrak{q}$ be a prime of $E$ above $\mathfrak{p}$, and let $\pi_E$ be the base-change lift of $\pi$ to $PGL_{n,E}$. Let $\operatorname{pr}\colon \operatorname{Hom}_{\operatorname{ct}}(E_{\mathfrak{q}}^{\times},\Omega)\to\operatorname{Hom}_{\operatorname{ct}}(F_{\mathfrak{p}}^{\times},\Omega)$ be the canonical projection. Base-change compatibility conjecture. Under these assumptions,

$$\mathcal{L}_{i}^{(0)}(\pi_E,\mathfrak{q})^{{\mathbbm 1}}=\operatorname{pr}^{-1}\bigl(\mathcal{L}_{i}^{(0)}(\pi,\mathfrak{p})^{{\mathbbm 1}}\bigr).$$

This predicts compatibility of automorphic $\mathcal{L}$-invariants with solvable automorphic base change; the preceding Galois-equivariance results explain why the invariant over $E$ should descend to $F$.