Assume $f(z)$ is analytic in the tube domain $\mathbb R^n\oplus iC$, where $C\subset \mathbb R^n$ is a convex cone. Under the assumption $|f(x+iy)|\leq 1/|y|^k$, we know by a Theorem of Martineau (see also Hormander, volume 1, Theorem 3.1.15) that the limit $\lim_{y\to 0, y\in C} f(x+iy)$ exists as a tempered distribution $f(x)$ on $\mathbb R^n$, uniformly in proper cones $y\in C'\subset C$. The convergence is in the weak topology, and in fact in the strong topology on the space of tempered distributions of fixed order $k$.

Question: Let $\Gamma\subset T^*\mathbb R^n$ be the wave front set of $f(x)$. Is it true that $f(x+iy)\to f(x)$ also in the Hormander topology $C^{-\infty}_\Gamma$? If not true in general, can some conditions be given on $f$ that would ensure such convergence?