The principal finite-bundle conjecture for the PCVP

Let $\Gamma$ be a finite group and $(E,X,\pi)$ be a principal $\Gamma$-bundle, where $(X,\widetilde{\mathbb{Z}}_X)$ is a ringed, connected, compact, Hausdorff, and PCVP space. Suppose that $X$ admits a covering by admissible subspaces. Principal finite-bundle conjecture. Under these hypotheses, the total space $E$ has the PCVP. This conjecture proposes extending the result for products with finite discrete spaces to principal finite bundles. Its status is not resolved in the supplied text.