The Gamma-conjecture III on exponential-type quantum connections

Let $X$ be a smooth projective variety, let $\tau\in H^*(X)$ be a point where the quantum connection is of exponential type, and let $\mathsf{C}$ index the exponential factors. Let $\mathcal{V}_u$ be the summands of the sectorial decomposition of flat sections, and let $\mathfrak{s}$ be the Gamma-integral-structure map. Gamma-conjecture III. There exists a decomposition

$$K(X)=\bigoplus_{u\in\mathsf{C}}V_u^\phi$$

such that $\mathcal{V}_u=\mathfrak{s}(V_u^\phi)\otimes\mathbb{C}$, with the lift $\phi$ determining the sector and with $V_u^{\phi+2\pi}=V_u^\phi\otimes\omega_X[n]$. This conjecture lifts the sectorial semiorthogonal decomposition from flat sections to the topological $K$-group lattice. In the semisimple case, the source notes that each lattice is expected to be isomorphic to $\mathbb{Z}$ and that its Euler form is unimodular; Gamma-conjecture II adds the stronger self-pairing condition $1$.