Extension of the family-of-representations theorem over locally closed special-fiber subschemes

Let $X$ be a reduced adic space locally of finite type over $\mathbb{Q}_p$, and let $(\mathcal{N},\Phi)$ be a family of étale $\varphi$-modules, or an étale $(\varphi,\Gamma)$-module, on $X$. The theorem and its corresponding corollary are formulated for the tube of a closed point $x_0$ in the special fiber of a formal model of $X$.

Extension conjecture. The claim of the theorem and the corollary also holds true if one replaces $x_0$ by a (locally) closed subscheme of the special fiber over which there exists a Galois representation that is locally associated with the reduction of an étale lattice.

This would extend the local association between étale $(\varphi,\Gamma)$-modules and families of Galois representations from tubes of closed points to suitable locally closed subschemes. The parser supplies no evidence that this statement has been resolved.