The conjecture relating $B(\chi,[1,b])$ to the $p$-adic local Langlands correspondence

For data $(\chi,[1,b])$, let $B(\chi,[1,b])$ be the admissible Banach representation constructed in the paper, and let $V_{\chi,[1,b]}$ be the associated two-dimensional Galois representation. Write $\Pi(V_{\chi,[1,b]}^\vee)$ for the Banach space representation of $\mathrm{GL}_2(\mathbb{Q}_p)$ attached to $V_{\chi,[1,b]}^\vee$ by the $p$-adic local Langlands correspondence.

Local Langlands conjecture for $B(\chi,[1,b])$. Up to a twist by a character, $B(\chi,[1,b])$ is isomorphic to $\Pi(V_{\chi,[1,b]}^\vee)$ as a Banach space representation of $\mathrm{GL}_2(\mathbb{Q}_p)$.

The claim proposes that the explicitly constructed completion $B(\chi,[1,b])$ agrees, up to character twist, with the representation predicted by the $p$-adic local Langlands correspondence. The source does not establish this isomorphism, so its resolution is left open here.