Small-radius convexity conjecture for Cassinian balls in the upper half-plane

Let $\mathbb{H}^2=\{z\in\mathbb{C}:\operatorname{Im}z>0\}$, let $\delta_{\mathbb{H}^2}(x)$ denote the Euclidean distance from $x$ to the boundary, and let $B_{c_{\mathbb{H}^2}}(x,r)$ be the Cassinian ball of radius $r$. Small-radius convexity conjecture. There exists $a>0$ such that $B_{c_{\mathbb{H}^2}}(x,r)$ is convex for all $x\in\mathbb{H}^2$ and $r\in(0,a\delta_{\mathbb{H}^2}(x)]$. Computer experiments suggest that this holds for $a\approx0.85$, but no proof is given.