Breuil–Schneider conjecture for generalized Robba-ring parametrizations

Let $X$ be as in the paper, let $n\geq 2$, let $G(X)$ be a reductive group, and let $\Pi_X[\log(1+T)^{1/n}]$ be the generalized Robba ring in the imperfect setting. Let $(\varphi,\Gamma_n)$ denote the indicated Frobenius and covering-group structure, and let $G^\mathrm{Lan}(\overline{T})$ be the Langlands dual object. Breuil–Schneider conjecture. In the $z$-adic situation, there exists a condensed parametrization of a $(\varphi,\Gamma_n)$-$G^\mathrm{Lan}(\overline{T})$-bundle over $\Pi_X[\log(1+T)^{1/n}]$ for generalized $p$-adic Banach representations of a $z$-adic reductive group $G(X)$; in the $p$-adic situation, the analogous parametrization exists for generalized $p$-adic Banach representations of a $p$-adic reductive group $G(X)$. The conjecture is a proposed Robba-ring version of the Breuil–Schneider consideration, and the source does not establish either assertion.