Vanishing conjecture for tempered holomorphic functions on relatively compact subanalytic pseudo-convex opens

Let $X=\mathbb{C}^n$, and let $U$ be a relatively compact subanalytic pseudo-convex open subset of $X$. The derived global sections of the sheaf of tempered holomorphic functions, $\mathscr{O}^{\mspace{2.5mu}{\mathrm t}}_{{X_{\sa}}}$, are objects of $\mathsf{D}^{\mathrm{b}}(\mathbb{C})$. Vanishing conjecture.

$$\mathrm{R}\Gamma(U;\mathscr{O}^{\mspace{2.5mu}{\mathrm t}}_{{X_{\sa}}})$$

is concentrated in degree $0$. The statement is presented as known to some specialists, but the source gives no reference and does not label it as a theorem; its status therefore remains open.