The $p$-adic intersection cohomology conjecture for rigid spaces

Let $K/\mathbf{Q}_p$ be a finite extension, let $G_K$ be its absolute Galois group, and let $X$ be a rigid space proper over $K$. Write $IH^i(X_C,\mathbf{Q}_p)$ for its $p$-adic intersection cohomology. If $j:U\to X$ is a smooth Zariski-open subset and $\mathbf{L}$ is a de Rham $\mathbf{Z}_p$-local system on $U$, write $\mathrm{IC}(\mathbf{L}[\dim(X)])$ for the corresponding intersection complex. $p$-adic intersection cohomology conjecture. Each $IH^i(X_C,\mathbf{Q}_p)$ is a de Rham $G_K$-representation; moreover, assuming the conjecture on intersection cohomology for Zariski-compactifiable spaces, this should hold for any Zariski-compactifiable rigid space. In addition, $H^*(X_{\overline{K}},\mathrm{IC}(\mathbf{L}[\dim(X)]))$ is de Rham. The first assertion is known in the algebraic case using the decomposition theorem, while the general rigid-analytic statement remains conjectural.