Weak admissibility conjecture for exponentially twisted cohomology

Let $f$ and its Teichmüller lift $\hat{f}$ be nondegenerate Laurent polynomials with $\dim\Delta(f)=n$. Let

$$V_{\mathrm{rig}}=H^{n}_{\mathrm{rig}}(\mathbb{T}^{n}_{\mathbb{F}_{q}}/K_{1},f^{\ast}\mathcal{L}_{\pi}),$$

with Frobenius structure $\phi_f$, and let

$$V_{\mathrm{dR}}=H^{n}_{\mathrm{dR}}(\mathbb{T}^{n}_{K_{1}},\nabla_{\widehat{F}}),$$

with irregular Hodge filtration $F^{\ast}_{\mathrm{irr}}$. Here $\iota_{\widehat{F}}:V_{\mathrm{dR}}\to V_{\mathrm{rig}}$ is the specialization map, $K_1=\operatorname{Frac}(\mathrm{W}(\mathbb{F}_q))(\zeta_p)$, and $\widehat{F}=\pi\hat f$.

Weak admissibility conjecture. The specialization map $\iota_{\widehat{F}}$ is an isomorphism, and

$$((V_{\mathrm{rig}},\phi_f),(V_{\mathrm{dR}},F^{\ast}_{\mathrm{irr}}),\iota_{\widehat{F}})\in\mathbf{M\overline{F}}{}^{\Phi}_{K_1}$$

is weakly admissible.

This conjecture predicts that the Frobenius structure and irregular Hodge filtration attached to exponentially twisted cohomology satisfy the compatibility required for weak admissibility. The source states that the expectation is proved under some assumptions using the theory of Adolphson and Sperber, while the full claim in the stated generality is not established here.