Connectedness conjecture for the crystalline deformation-ring fibre

Let $\mathbb{F}$ be a finite field of characteristic $p$, let $K$ be a finite extension of $\mathbb{Q}_p$ with absolute Galois group $G_K$, and let $V_{\mathbb{F}}$ be an absolutely irreducible finite-dimensional $\mathbb{F}$-representation of $G_K$. Assume that $\mathbb{F}$ is sufficiently large and that

$$V_{\mathbb{F}}=\operatorname{Ind}_{G_L}^{G_K}W_{\mathbb{F}},$$

where $L/K$ is unramified, $W_{\mathbb{F}}$ is one-dimensional, and the residue field $l$ of $L$ embeds into $\mathbb{F}$. Let $R=R^\square_{V_{\mathbb{F}}}$, let $\mathcal{L}=\mathcal{L}^{\leq p}_{R,\operatorname{crys}}$, and let $\mathcal{L}_{\mathbb{F}}=\mathcal{L}\otimes_R\mathbb{F}$ be its fibre over the closed point of $\operatorname{Spec}R$. Let $f_*\colon\mathcal{L}^{\leq p}(W_{\mathbb{F}})\to\mathcal{L}^{\leq p}(V_{\mathbb{F}})$ be the map induced by induction. Connectedness conjecture. Every closed point of $\mathcal{L}_{\mathbb{F}}$ lies in the same connected component as a closed point arising from $f_*\mathfrak{N}$ for some $\mathfrak{N}\in\mathcal{L}^{\leq p}(W_{\mathbb{F}})$. This conjectural connectedness statement would strengthen the paper's potential diagonalizability result by controlling the connected components of the crystalline deformation-ring fibre; its resolution is not specified in the source.