A sharp $p$-subadditive bound for the $l_p$ Hausdorff distance from convex hull

We study the $\ell_p$ Hausdorff distance from convex hull, which for a compact set $A\subset \mathbb{R}^n$ is defined by
\begin{align*}
d^{(\ell_p)}(A):=\sup_{x\in \text{conv}(A)}\inf_{a\in A}\|x-a\|_p.
\end{align*}
In the planar case $n=2$, we study the problem of finding the optimal constant $C_p$ such that
\begin{align*}
d^{(\ell_p)}(A+B)^p\leq C_p\left(d^{(\ell_p)}(A)^p+d^{(\ell_p)}(B)^p\right)
\end{align*}
for all nonempty compact $A,B\subset\mathbb{R}^2$. We resolve this question, proving that
\begin{align*}
C_p=\max\{1,2^{p-2}\}. \end{align*}