Coherent-sheaf realization conjecture for smooth representations

Let $K$ be a nonarchimedean local field, let $d$ be a positive integer, and let $]_{ll]}$ be the coefficient ring. Let $L]_d(K)$-representations mean representations on $]_{ll]}$-modules, and let $im]_{Q}]_Q$ be the Ind-algebraic stack appearing in the conjectural enhancement of the stated theorem. Coherent-sheaf realization conjecture. The category of $L]_d(K)$-representations on $]_{ll]}$-modules admits a fully faithful embedding into the category of coherent sheaves on $im]_{Q}]_Q$, compatibly with the identification of Theorem~$m]$. This is presented as a rough and speculative form of a $p$-adic local Langlands correspondence, motivated by analogous conjectures for $ll]\ne p$. No resolution is given in the supplied text.