Unipotent representation conjecture for graded Langlands parameters

Let $G$ be a connected reductive group, let $G_\eta$ range over its pure inner forms, and let $\mathbb{L}^u_{\underline q}$ be the graded unipotent Langlands parameter space. Write $D_{f.g.}(\check G_\eta)^u$ and $D(\check G_\eta)^u$ for the indicated unipotent representation categories, and let ${\operatorname{Coh}}(\mathbb{L}^u_{\underline q})^{S^1}$ and $\QC^!(\mathbb{L}^u_{\underline q})^{\Tate}$ denote cyclic and periodic cyclic sheaves. Unipotent representation conjecture. There should be a full embedding

$$\bigoplus_\eta D_{f.g.}(\check G_\eta)^u\otimes_k k[u] \subset \operatorname{Coh}(\mathbb{L}^u_{\underline q})^{S^1},$$

which becomes an equivalence after inverting $u$:

$$\bigoplus_\eta D(\check G_\eta)^u\otimes_k k[u,u^{-1}]\simeq \QC^!(\mathbb{L}^u_{\underline q})^{\Tate}.$$

The expectation is that cyclic deformation interpolates between all isocrystals and pure inner forms, identifying periodic cyclic sheaves with unipotent representations of pure inner forms; the statement is presented as a proposal in the source.