Conjecture on the explicit irregular Hodge module structure of Frenkel–Gross connections

Let $\check G$ be the Langlands dual group, let $V\in\operatorname{Rep}(\check G)$, and let $\nabla_{\check G}^{\mathrm{H}}(V)$ be the rescalable integrable mixed twistor module associated with the Frenkel–Gross connection. Write

$$(\mathcal{M},\mathcal{M}',C)$$

for its $\mathcal{R}$-triple, and let $\nabla_{\check G}^{\mathcal{R}}(V)$ denote the explicit integrable $\mathcal{R}$-module constructed from the cocharacter $\rho$. Explicit irregular Hodge module conjecture. For every $V\in\operatorname{Rep}(\check G)$, there is an isomorphism

$$\mathcal{M}\cong\nabla_{\check G}^{\mathcal{R}}(V)$$

of integrable $\mathcal{R}$-modules. Consequently, the irregular Hodge filtration of $\nabla_{\check G}^{\mathrm{H}}(V)$ is determined by $\rho$. This would identify the abstract irregular Hodge module structure with the explicit rescaling construction and would provide an alternative route to the main result; the conjecture is presented as open in the source.