No common unfolding for most pairs of doubly covered triangles

Prove or disprove that most pairs of doubly covered triangles, specifically those with rationally independent angles, have no common unfolding and therefore no 1-step refolding.

Background

Doubly covered triangles are convex polyhedra formed by gluing two congruent triangular faces. Whether distinct such shapes can share a common unfolding is central to 1-step refolding questions.

The cited conjecture asserts that most such pairs lack a common unfolding, providing evidence toward the necessity of at least two refolding steps in some cases.