The exponential type conjecture for quantum t-connections

Let $(X,comega)$ be a closed monotone symplectic manifold of dimension $2n$, with $[comega]=c_1(X)$. Let $t$ be a formal variable of degree $2$, and let

$$\nabla^{QH}_{\frac{\partial}{\partial t}}:=\frac{\partial}{\partial t}+\frac{\mu}{t}-\frac{c_1\star}{t^2}$$

be the quantum $t$-connection on $H^*(X;\mathbb{C})[[t]]$, where $\mu|_{H^k(X)}=\frac{k-n}{2}$ and $\star$ is the small quantum cup product. The exponential type conjecture. The quantum $t$-connection $(H^*(X;\mathbb{C})((t)),\nabla^{QH}_{\frac{\partial}{\partial t}})$ admits a finite direct sum decomposition

$$\nabla^{QH}_{\frac{\partial}{\partial t}}=\bigoplus_{\lambda}(\mathbb{C}((t)),\frac{\partial}{\partial t}-\frac{\lambda}{t^2})\otimes \nabla^{reg,\lambda}_{\frac{\partial}{\partial t}},$$

where $\lambda\in\mathbb{C}$ and each $\nabla^{reg,\lambda}_{\frac{\partial}{\partial t}}$ is gauge equivalent to a connection with simple poles at $t=0$ and monodromies given by roots of unity. Equivalently, the quantum connection has unramified exponential type and quasi-unipotent regularized monodromy. This conjecture concerns the singularity and monodromy structure of quantum connections; the source paper proves it for closed monotone symplectic manifolds, while the general conjecture is attributed to Katzarkov–Kontsevich–Pantev and Galkin–Golyshev–Iritani.