Contracted-triangulation conjecture for closed simply-connected PL 4-manifolds

Let $M$ be a closed simply-connected PL $4$-manifold. A contracted triangulation is a triangulation satisfying the hypothesis of Theorem (b). Contracted-triangulation conjecture. Every closed simply-connected PL $4$-manifold admits a contracted triangulation satisfying that hypothesis.

This is proposed as a weaker conjecture than the additivity conjecture for regular genus, while retaining the stated implications for exotic structures, the Poincaré conjecture, and special handlebody decompositions. Its status remains open.