Transport-distance finiteness for uniformly hyperuniform random measures

Let $p\geq -d+2$. Suppose that $\Phi$ is an invariant locally absolutely square-integrable random (nonnegative) measure with intensity $\gamma$.

Transport-distance finiteness conjecture. If $\Phi$ is $(p+\vartheta)$-uniform for some $\vartheta>0$, then

$$d_{d+p}(\Phi, \gamma\lambda_d)<\infty.$$

This conjecture proposes that sufficiently strong uniformity of the random measure ensures finiteness of the corresponding transport distance to the stationary measure $\gamma\lambda_d$. The supplied text does not state whether the claim is known or resolved.