Let $(X^{4},g)$ be a complete Riemannian $4$-manifold which is $\delta$-asymptotically Calabi for some $\delta>0$ and which has $\Ric_{g}\geq 0$. Then there exists an $\ell_{0}\in(0,1)$ depending on $(X^{4},g)$ such that if $u$ is a harmonic function on $(X^{4},g)$ with $u=O(e^{\ell_{0}z})$ as $z\to\infty$, then $u$ is a constant.