Sufficiency of the minimal proper-interval completion characterization

Let $G$ be an interval graph, and let $H$ be a completion of $G$ to a proper interval graph. The cited theorem gives a necessary “only if” condition for $H$ to be minimal.

Minimal proper-interval completion conjecture. The only-if condition of that theorem is also sufficient. Moreover, if this condition is sufficient, then the problem of finding a minimal completion to a proper interval graph, when the input is an interval graph, has polynomial complexity.

This conjecture concerns the characterization and complexity of minimal completions from interval graphs to proper interval graphs. The source does not provide evidence resolving it, so its status remains open.