The non-ordinary base-change classification conjecture

Let $\mathcal E_{\mathrm{par}}$ be the parallel-weight eigenvariety, let $\mathcal C_M$ denote the relevant Coleman–Mazur eigencurve at level parameter $M$, and let $[\varphi]$ denote twisting by a finite-order Hecke character $\varphi$ of $K$ of conductor prime to $p$. The non-ordinary base-change classification conjecture. For every non-ordinary irreducible component $\mathcal E'$ of $\mathcal E_{\mathrm{par}}$, there exist an integer $M$, an irreducible component $\mathcal C'_M$ of $\mathcal C_M$, and such a character $\varphi$ such that

$$\mathcal E'=[\varphi]\circ\mathrm{BC}(\mathcal C'_M).$$

The source says that this generalisation of the Calegari–Mazur conjecture would imply the preceding $\Sigma$-smoothness assertion; it remains conjectural there.