Guarantee of LADC for non-trivial radial updates

Let $X$ be a radial Riemannian manifold with stationary probability distribution $p(x)\propto{\mathrm{e}^{-V(x)}}$. Suppose an update scheme changes the radial component in a non-trivial manner, and let $X_0$ be a compact subset of $X$. Guarantee of LADC. The update scheme satisfies the LADC for the potential $V$ on $X\setminus X_0$. The statement is presented as a guarantee for radial-update Monte Carlo schemes; the supplied text does not establish whether it has been proved or remains open.