K-theory continuation conjecture for global quantum D-modules

Let $\mathcal{X}_1$ and $\mathcal{X}_2$ be K-equivalent spaces connected by a global quantum $D$-module, and let $\gamma$ be a path in the Kähler moduli space from a point of the cusp neighborhood $V_1$ to one of $V_2$. Let $P_\gamma$ be the induced analytic-continuation map on flat sections, and let $K(\mathcal{X}_i)$ denote the corresponding $K$-groups. K-theory continuation conjecture. For each path $\gamma$, there exists an isomorphism

$$\mathbb{U}_{K,\gamma}\colon K(\mathcal{X}_1)\to K(\mathcal{X}_2)$$

which induces $P_\gamma$ through the K-group framing and preserves the Mukai pairing:

$$\chi\bigl(\mathbb{U}_{K,\gamma}(V_1)\otimes\mathbb{U}_{K,\gamma}(V_2)^\vee\bigr)=\chi(V_1\otimes V_2^\vee).$$

The conjecture would determine the relationship between the two quantum $D$-modules, apart from analytic continuation. It is proposed in the paper and remains unproved in general.