Temperate holomorphic solutions and regularity of holonomic D-modules on curves

Let $X$ be a complex manifold. In “Microlocal study of Ind-sheaves I: microsupport and regularity”, M. Kashiwara e P. Schapira made the conjecture that a holonomic D-module $\text{shm}$ is regular holonomic if and only if $R\mathcal{I}{hom}_{{\beta }_X{\text{shd}}_X}({\beta }_X\text{shm},{\text{sho}}_X^t)$ is regular (in the sense of “Microlocal study of Ind-sheaves I: microsupport and regularity”), the “only if” part of this conjecture following immediately from “Microlocal study of Ind-sheaves I: microsupport and regularity”. Our aim is to prove this conjecture in dimension one.