Weakly real-constructible ind-sheaves are closed under ind-internal Hom and tensor product

Let $X$ be a real analytic manifold, let $F\in D^b_{{\rm w-}\mathbb{R}\text{-c}}(\mathrm{I}[k_X])$, and let $G\in D^b_{{\rm w-}\mathbb{R}\text{-c}}(k_X)$. Closure conjecture. The objects $R\mathcal{I}\operatorname{hom}(G,F)$ and $G\otimes F$ belong to $D^b_{{\rm w-}\mathbb{R}\text{-c}}(\mathrm{I}[k_X])$. This is the closure property underlying the theory of weakly real-constructible ind-sheaves; the supplied text gives no evidence of a proof or disproof.