Gehrmann's conjecture on automorphic $\mathcal{L}$-invariant spaces

Let $\pi$ be the automorphic representation and let $c\in\{1,\dots,n-1\}$, $0\le r\le\ell$, and $\varepsilon\in\mathcal{E}_\pi$. Let $\mathcal{L}_c^{r,\varepsilon}(\pi)$ be the subspace of $\operatorname{Hom}_{\operatorname{cts}}(\mathbb{Q}_p^\times,L)$ defined by the vanishing of the relevant cup-product pairing. Gehrmann's conjecture. The following assertions hold: (i) $\mathcal{L}_c^{r,\varepsilon}(\pi)$ has codimension exactly $1$, equivalently dimension exactly $1$, for every $c\in\{1,\dots,n-1\}$, $0\le r\le\ell$, and $\varepsilon\in\mathcal{E}_\pi$; (ii) $\mathcal{L}_c^{r,\varepsilon}(\pi)$ is independent of both $r$ and $\varepsilon\in\mathcal{E}_\pi$. This predicts that the automorphic $\mathcal{L}$-invariant space is a single line, independent of the cohomological degree and admissible sign. Part (i) is known in the bottom and top degrees $r=0$ and $r=\ell$ by the proposition immediately preceding the conjecture; the supplied text gives no resolution for the remaining cases or for the independence assertion.