Capacitary domination and Fefferman--Phong inequalities for measure potentials on $2$-PI spaces

Let $(X,d,ω)$ be an unbounded complete doubling metric measure space supporting a weak $(1,2)$-Poincaré inequality, and let $π$ be a positive Radon measure. For a ball $B=B(x,R)$, suppose that for some $δ>0$ and $C_{\mathrm{gr}}>0$, \[ π(B(y,r))\le C_{\mathrm{gr}} \left(\frac rR\right)^δ\frac{ω(