The reconstruction conjecture for holonomic systems

Let $X$ be a complex analytic manifold and let $\boldsymbol{P}$ denote the complex projective line. Consider the contravariant functors

$$\Phi\colon D(\mathcal D_X)\rightleftarrows D(\mathrm{I}\mathbb C_{X\times\boldsymbol{P}})\colon\Psi$$

obtained from tempered holomorphic solutions after tensoring with the exponential module associated with the affine coordinate on $\boldsymbol{P}$, where $\boldsymbol{P}$ is equipped with the projections from $X\times\boldsymbol{P}$. Reconstruction conjecture. The natural morphism of endofunctors

$$\operatorname{id}\longrightarrow\Psi\circ\Phi$$

is an isomorphism on $D_{\mathrm{hol}}(\mathcal D_X)$, and the restriction

$$\Phi|_{D_{\mathrm{hol}}(\mathcal D_X)}\colon D_{\mathrm{hol}}(\mathcal D_X)\longrightarrow D(\mathrm{I}\mathbb C_{X\times\boldsymbol{P}})$$

is fully faithful. This would provide a Riemann–Hilbert correspondence for holonomic systems, extending the classical correspondence from regular holonomic systems to arbitrary holonomic systems and recovering a holonomic system from its ind-constructible complex of tempered solutions.