A subtlety is that the local regularity theory of real Monge-Ampère equation allows for Hausdorff codimension $1+\epsilon$ singularities. This means the codimension two singularity prediction in the Kontsevich-Soibelman conjecture cannot follow simply from the above argument. One must find a more global argument on $Sk(X)$, not just on the interior of its $n$-dimensional faces.