Convolution of harmonic function $h$ constant implies that $h$ is constant?

Question

Let $C\subseteq \mathbb{R}^d$ be a compact set and $\rho\in L^1(\mathbb{R}^d,[0,+\infty))$ a function whose essential support is $C$. For some $R\in (0,+\infty)$, let $U:=C+B(0,R)$ and $h:U\to \mathbb{R}$ a harmonic function. It is well-known that the convolution $$x\mapsto \int_{C\subseteq \mathbb{R}^d}dy\, \rho(y)h(y+x)$$ (with domain $B(0,R)$) is harmonic too. But if that convolution is constant and $\rho$ is non-zero, does it mean that $h$ was constant to begin with?