Conjecture on degeneration of singular AC metrics to smooth AC metrics

Let $E\to B$ be a rank-$r$ vector bundle, let $p:\mathbb{P}(E)\to B$ be its projectivization, and let $L$ be the line bundle such that the total space $E\otimes L$ is a crepant resolution of the cone over $\mathbb{P}(E)$. The $r$-th root of $K_{\mathbb{P}(E)}$ carries singular asymptotically conical (AC) metrics, while $E\otimes L$ carries smooth AC metrics.

Metric contraction conjecture. The singular AC metrics on the $r$-th root of $K_{\mathbb{P}(E)}$ contract along one-parameter families to the smooth AC metrics on $E\otimes L$ by shrinking the fibers of $p$ in the zero section.

This predicts a geometric transition from the branched-cover metrics on the singular space to the smooth metrics on its small crepant resolution. The source gives no resolution of this claim.