The canonical splitting conjecture for the quantum connection

Let $M$ be a closed monotone symplectic manifold, let $R$ be an integral domain, and write its quantum cohomology as $H^*(M)[q]$, with quantum product $\ast_q$. Suppose that, after inverting $q$, there are mutually orthogonal idempotents $e_1,\dots,e_m\text{ in }H^*(M)[q^{\pm 1}]$ of degree zero satisfying

1=e_1+\cdots+e_m,\qquad e_i\ast_q e_j=\begin{cases}e_i&i=j,\\0&i\ne j.end{cases}

Let $t$ be a formal variable of degree $2$, and define the quantum connection on $H^*(M)[q,t]$ by

$$\nabla_{tq\partial_q}x=tq\partial_qx+c_1(M)\ast_qx.$$

Canonical splitting conjecture. Given these idempotents, the space $H^*(M)[q^{\pm 1}][[t]]$ carries a canonical splitting into graded $R[q^{\pm 1}][[t]]$-modules, invariant under $\nabla_{tq\partial_q}$, whose $t=0$ reduction agrees with the splitting of $H^*(M)[q^{\pm 1}]$ given by quantum product with the idempotents.

This expectation arises from the anticipated compatibility of Fukaya-categorical structures with the completed quantum connection. The statement asks for a canonical extension of the idempotent splitting from quantum cohomology to the $t$-adic completion, preserving the quantum connection; no resolution status is supplied here.