Representability conjecture for triangulordinary filtrations

Let $\mathscr{X}/E$ be a reduced, separated rigid analytic space, and let $\mathscr{D}$ be a locally free sheaf of $(\varphi,\Gamma_K)$-modules over $\mathcal{O}_{\mathscr{X}} {\widehat{\otimes}} B^\dag_{\operatorname{rig},K}$, of rank $d$. Let $0<c<d$ be an integer. For an $\mathscr{X}$-space $f\colon\mathscr{U}\to\mathscr{X}$, consider the collection of $(\varphi,\Gamma_K)$-stable $\mathcal{O}_{\mathscr{U}} {\widehat{\otimes}} B^\dag_{\operatorname{rig},K}$-local direct summands of $f^*\mathscr{D}$ of rank $c$. Representability conjecture. The functor assigning this collection to $f$ is representable by a locally finite type morphism

$$p_c\colon\mathscr{X}(c)\to\mathscr{X}.$$

For each $x\in\mathscr{X}$, the fiber $\mathscr{X}(c)_x$ is a finite union of quasiprojective flag varieties over the residue field $E(x)$. This would provide a geometric parameter space for triangulordinary filtrations in families; the statement is presented as a conjectural interpolation result, and no resolution is supplied here.