Comparison conjecture for arithmetic $\mathcal{D}$-module and rigid cohomology

Let $X_\eta$ be the generic fibre of the strictly semi-stable scheme considered in the paper, let $\mathcal{E}^{\dagger}_K$ be the bounded Robba-ring coefficient field, and write $\mathbb{R}\Gamma_{\mathcal{D}}$ and $\mathbb{R}\Gamma_{\mathrm{rig}}$ for arithmetic $\mathcal{D}$-module and rigid cohomology. Comparison conjecture. There should be a canonical isomorphism

$$\mathbb{R}\Gamma_{\mathcal{D}}(X_\eta/\mathcal{E}^{\dagger}_K)\simeq\mathbb{R}\Gamma_{\mathrm{rig}}(X_\eta/\mathcal{E}^{\dagger}_K)$$

such that the induced isomorphism

$$H^i_{\mathcal{D}}(X_\eta/\mathcal{E}^{\dagger}_K)\simeq H^i_{\mathrm{rig}}(X_\eta/\mathcal{E}^{\dagger}_K)$$

is compatible with the structures of $\nabla$-modules and with pullbacks. This is motivated by the comparison theorem cited in the paper and by Lazda–Pál's rigid cohomology over bounded Robba rings; the source presents it as expected rather than established.