HMS symmetries of toric boundary divisors

Let $X$ be a projective crepant resolution of a Gorenstein affine toric variety and let $((\mathbb{C}^*)^k,f)$ be the LG-model which is the Hori-Vafa mirror dual of $X$. Let ${D}$ be a generic fiber of $f$ equipped with the restriction of the standard Liouville form on $(\mathbb{C}^*)^k$. Let $\mathcal{K}_A$ be the so-called "stringy K\"ahler moduli space" of $X$. We show that $\pi_1(\mathcal{K}_A)$ acts on the wrapped Fukaya category of $D$. Using results by Gammage - Shende and Zhou, this result implies that $\pi_1(\mathcal{K}_A)$ acts on $D^b(\operatorname{coh}(\partial X))$ where $\partial X$ is the toric boundary divisor of $X$. We show that the induced action of $\pi_1(\mathcal{K}_A)$ on $K_0(\operatorname{coh}(\partial X))$ may be extended in a natural way to an action on $K_0(X)$ which corresponds to a GKZ system.