Steingart's surjectivity conjecture for Lubin–Tate Iwasawa cohomology

Let $L$ be a finite extension of $\mathbb{Q}_p$, let $S$ be a representation of $G_L$, and let $S^*(1)$ be $L$-analytic. Write $\tau=\chi_{\mathrm{LT}}^{-1}\chi_{\mathrm{cyc}}$, let $\Lambda$ be the Iwasawa algebra, let $\Lambda_\infty^L$ be the $L$-analytic distribution algebra, and let $D^{\dagger}_{\operatorname{rig}}(S(\tau^{-1}))$ be the associated Robba-ring $(\varphi,\Gamma)$-module with operator $\Psi$. The comparison map is

$$\operatorname{comp}:R\Gamma_{\mathrm{Iw}}(L_\infty/L,S)\otimes_\Lambda\Lambda_\infty^L\longrightarrow D^{\dagger}_{\operatorname{rig}}(S(\tau^{-1}))[\Psi-1].$$

Steingart's surjectivity conjecture. The map $\operatorname{comp}$ is surjective after taking $H^1$, namely

$$H^1_{\mathrm{Iw}}(L_\infty/L,S)\otimes_\Lambda\Lambda_\infty^L\twoheadrightarrow D^{\dagger}_{\mathrm{rig}}(S(\tau^{-1}))^{\Psi=1}.$$

In the classical case $L=\mathbb{Q}_p$, the comparison map is known to be a quasi-isomorphism, whereas for $L\ne\mathbb{Q}_p$ it is not a quasi-isomorphism in general. The conjecture asserts that surjectivity nevertheless survives on $H^1$; proving it in this generality remains open.