Converse of (*) for small and rigid geometric transitions

A geometric transition is abbreviated g.t.; a geometric transition is small and rigid when it has those properties. A factorization of a geometric transition $T$ is an expression

$$T=T_2\circ T_1.$$

A conifold transition is a geometric transition through conifold singularities.

Converse of (*). Every small and rigid geometric transition $T$ admits a factorization $T=T_2\circ T_1$ whose first factor $T_1$ is a conifold transition.

This conjecture asks for the converse of the preceding rigidity statement and would give a conifold factor in every small rigid geometric transition. The supplied text gives no resolution status.