$\pi(z)\otimes\delta(r-r_0) = \sqrt{r_0^2 - z^2}\,\delta\left(r-\sqrt{r_0^2 - z^2}\right)$?

Question

I'm trying to solve for $\boldsymbol{\pi}(z)$ in a purported equivalence relation:

$\boldsymbol{\pi}(z)\otimes\delta(r-r_0) = \sqrt{r_0^2 - z^2}\,\delta\left(r-\sqrt{r_0^2 - z^2}\right)$

where $\boldsymbol{\pi}(z)$ is an "xy-plane" (in some sense) convolved with a sphere $\delta(r-r_0)$ of radius $r_0$.

Is there any solution for $\boldsymbol{\pi}(z)$ that would make this identity hold? If so, what exactly is that interpretation?