Uniqueness conjecture for the triangulordinary filtration on the eigencurve

Let $\mathscr{E}^0$ be the semistable, distinct-eigenvalue locus of the eigencurve, let $\mathscr{V}^0$ be its rank-$2$ family of Galois representations, and set

$$\mathscr{D}^0=\mathbf{D}^\dag_{\operatorname{rig}}(\mathscr{V}^0|_{G_p}).$$

For each $x\in\mathscr{E}^{0,\operatorname{alg}}$, write $F_x^1$ for the canonical rank-$1$ step in the triangulordinary filtration of $\mathbf{D}^\dag_{\operatorname{rig}}(V_{f_x}^{\operatorname{hom}}|_{G_p})$. Eigencurve triangulordinary-filtration conjecture. There exists a unique filtration

$$\mathscr{D}^0\supsetneq F^1\supsetneq0$$

by a $(\varphi,\Gamma_{\mathbb{Q}_p})$-stable locally $\mathcal{O}_{\mathscr{E}^0}{\widehat{\otimes}}B^\dag_{\operatorname{rig},\mathbb{Q}_p}$-direct summand $F^1$ of rank $1$, such that for every $x\in\mathscr{E}^{0,\operatorname{alg}}$, $(F^1)_x=F_x^1$ under the stated identification of fibers. This predicts that the pointwise triangulordinary filtrations on classical eigencurve points interpolate uniquely over the whole eigencurve; the source gives no resolution.