The mixed TERP-structure correspondence conjecture

Let $(H,\nabla,H'_{\mathds R},P)$ be a TERP-structure. It requires no ramification when its formal decomposition can be made without ramification; it is a mixed TERP-structure when real structure and Stokes structure are compatible and its regular singular pieces induce polarized mixed Hodge structures. It induces a nilpotent orbit when $\pi_r^*(H,\nabla,H'_{\mathds R},P)$ is a polarized pure TERP-structure for every $r\in{\mathds C}^*$ with $|r|\ll1$, where $\pi_r:{\mathds C}\to{\mathds C}$ is given by $z\mapsto z\cdot r$. The mixed TERP-structure correspondence conjecture. A TERP-structure which does not require a ramification is a mixed TERP-structure if and only if it induces a nilpotent orbit. This generalizes the correspondence between nilpotent orbits of Hodge structures and polarized mixed Hodge structures to arbitrary TERP-structures. One implication is proved in the paper, while the converse remains open in the general irregular case.