Compatibility of the two constructions of the extension $\pi^\alpha(\rho_p)$

Let $\pi^\alpha(\rho_p)$ be the nonsplit extension attached to $\rho_p$ in the cited work, and let $\pi^\alpha(\operatorname{Fil}^{\rm \max}_\alpha(\rho_p))$ be the extension constructed from the maximal filtration line. Compatibility conjecture. With the preceding notation, one has

$$\pi^\alpha(\rho_p)\simeq\pi^\alpha(\operatorname{Fil}^{\rm \max}_\alpha(\rho_p)).$$

This conjecture asserts that the global construction and the filtration-theoretic local construction give the same extension; it is stated for arbitrary Hodge–Tate weights and remains open in the supplied text.