Non-commutative crystalline comparison conjecture

Let $K$ be a $p$-adic field with ring of integers $\mathcal{O}_K$, residue field $k$, and absolute Galois group $G_K$. Let $W$ be the Witt-vector ring and $B_{\operatorname{crys}}$ the crystalline period ring. For a smooth proper $\mathcal{O}_K$-linear category $\mathcal{T}$, write $\mathcal{T}_k$ and $\mathcal{T}_{\mathcal{C}}$ for its special and completed generic fibres.

Non-commutative crystalline comparison conjecture. There is an isomorphism of $B_{\operatorname{crys}}$-modules

$$\pi_i\operatorname{TP}(\mathcal{T}_k;\mathbb{Z}_p)\otimes_W B_{\operatorname{crys}}\simeq \pi_i L_{K(1)}K(\mathcal{T}_{\mathcal{C}})\otimes_{\mathbb{Z}_p}B_{\operatorname{crys}},$$

compatible with the $G_K$-action and Frobenius endomorphism. In particular, the $p$-adic representation $\pi_iL_{K(1)}K(\mathcal{T}_{\mathcal{C}})\otimes_{\mathbb{Z}_p}\mathbb{Q}_p$ is crystalline. This predicts a non-commutative analogue of the crystalline comparison theorem; the source gives no general proof, though related work establishes the commutative comparison theorem and motivates the conjecture.