Breuil–Schneider conjecture for condensed generalized $z$-adic Banach representations

Let $X$ be a finite extension of a local nonarchimedean field, let $G(X)$ be a $z$-adic reductive group, and let $\mathrm{Gal}_{X,2}$ denote the two-fold covering of the Galois group obtained by taking roots of cyclotomic characters up to order $2$. Let $T$ be sufficiently large, as in the source, and let $G^\mathrm{Lan}(\overline{T})$ be the corresponding Langlands dual object. Breuil–Schneider conjecture. There exists a condensed parametrization

$$\mathrm{Gal}_{X,2}\rightarrow G^\mathrm{Lan}(\overline{T})$$

for generalized $z$-adic Banach representations of $G(X)$. The author notes that the relevant generalized $z$-adic Banach representations are not yet defined in the paper; the conjecture proposes that such a definition and parametrization should exist, extending the Breuil–Schneider perspective to the $z$-adic setting.