Ganatra–Kuo–Li–Wu conjecture on proper sheaf kernels

Let $X$ and $Y$ be manifolds, let $[?]$ be conic subsets specifying the microsupported sheaf categories, and let $K \in \Sh_{-\Lambda \times \Sigma}(X \times Y)$ be a sheaf kernel. Write $\ssupp(K)$ for its microsupport and suppose that $K$ has perfect stalks. Ganatra–Kuo–Li–Wu conjecture. If $Y$ is compact and the map

$$\ssupp(K) \longrightarrow T^*Y$$

is proper, then convolution

$$(-) \ast K \colon \Sh_{\Lambda}(X) \longrightarrow \Sh_{\Sigma}(Y)$$

preserves compact objects. The paper proves this special case as a consequence of its main result, so the conjecture is resolved.