The local companion points conjecture

Let $K$ be the coefficient field, let $L$ be a finite extension of $\mathbb{Q}_p$, and let $\Sigma$ denote the relevant embeddings of $K$. Consider a point

$$z=(r,\delta_1,\ldots,\delta_n)\in \mathrm{Spf}(R^{\square}_{\overline r})^{\mathrm{ad}}_\eta(L)\times\mathcal{T}^n(L).$$

Assume that $r$ is regular with integral $\tau$-Hodge--Tate--Sen weights $\{h_{\tau,1}<\cdots<h_{\tau,n}\}$ for every $\tau\in\Sigma$. Let $x\in X(L)$ be the point associated with $r$, and let $\mathcal{S}(x)$ be the set of permutations indexing the components of $X$ that pass through $x$.

The local companion points conjecture. The following conditions are equivalent:

1. $z\in X_{\mathrm{tri}}(\overline r)(L)$.
2. $r$ is trianguline, with a triangulation having parameters $\delta'_1,\ldots,\delta'_n$, and there is a permutation $w\in\mathcal{S}(x)$ such that, for every $i\in\{1,\ldots,n\}$,

$$\delta_i=\delta'_i\prod_{\tau\in\Sigma}x_\tau^{h_{\tau,w_\tau(i)}-\mathrm{wt}_\tau(\delta'_i)}.$$

This conjecture is a local analogue of companion-points conjectures in the global setting and is implicit in work of Breuil--Hellmann--Schraen. It characterizes membership in the trianguline variety through triangulations whose parameters are related to the Hodge--Tate data by permutations arising from the geometry of the associated point. The statement is given for regular integral weights; a version for regular non-integral weights requires modified definitions of $X$ and of the point associated with $r$.