Wormhole conjecture for punctured W-surfaces

Let $X_1$ be a W-surface and let $X_2$ be the new W-surface obtained by the extremal P-resolution surgery, with $K_{X_1}^2=K_{X_2}^2$ and $\chi(\mathcal{O}_{X_1})=\chi(\mathcal{O}_{X_2})$. Suppose that the MMP is run on $X_2$.

Wormhole conjecture. The MMP on the new W-surface finishes in a minimal model and requires only flips; that is, both punctured W-surfaces live in the same moduli space.

This is the punctured-surface formulation of the wormhole idea: preserving the numerical invariants and using only flips should keep the two surfaces in one moduli space. The supplied text does not state whether this has been proved or disproved.