Mazzeo's weighted X-ray transform isomorphism conjecture

Let $(M,g)$ be a simple Riemannian manifold with boundary defining function $\rho$, and fix $\gamma>-1$. An $\alpha$-boundary defining function ($\alpha$-bdf) for $\partial_+SM$ is denoted by ${\mathbf t}$. Mazzeo's conjecture. There exists an $\alpha$-bdf ${\mathbf t}$ for $\partial_+SM$ such that the operator

$$I_0^\sharp {\mathbf t}^{-2\gamma-1} I_0 \rho^\gamma$$

is an isomorphism of $C^\infty(M)$. This conjecture predicts that a suitable boundary weight makes the weighted normal X-ray operator invertible on smooth functions; the supplied text does not state whether it has been proved or disproved.