Standardness conjecture for Berge-knot homotopy projective planes

Let $X$ be a homotopy $\mathbb{CP}^2$ constructed in Proposition~ from a Berge knot. Standardness conjecture. The manifold $X$ is diffeomorphic to the standard $\mathbb{CP}^2$. These constructions produce homotopy $\mathbb{CP}^2$'s with handle decompositions having only one $3$-handle, and the source presents the claim as an open conjecture.