The projective compactification no-retraction conjecture

Let $(\Gamma\cup X,\tau)$ be a projective $\Gamma$-compactification, where $X$ is non-trivial, meaning that it contains more than one point. A $\Gamma$-compactification is Dirac if it admits a $\Gamma$-equivariant retraction onto its boundary $X$.

Projective compactification no-retraction conjecture. Every projective $\Gamma$-compactification with non-trivial boundary $X$ is non-Dirac. In particular, there does not exist a retraction map onto $X$.

Projective compactifications include Gromov compactifications of hyperbolic groups. The conjecture is motivated by the belief that projectivity is the main obstruction to point-orbitality and is stated as holding in all cases known to the authors; its general status is not resolved in the supplied source.