Bounded excess-height conjecture for 3-sphere triangulations

Let $\mathscr{P}_1(S^3)$ be the graph whose nodes are one-vertex triangulations of the 3-sphere, arranged by level $n$, and let a simplification path be a path reducing the level to a minimal triangulation. The excess height of such a path is the maximum level reached above its starting level. Bounded excess-height conjecture. From any node at any level $n \geq 3$ of the graph $\mathscr{P}_1(S^3)$, there is a simplification path of excess height $\leq 2$. If true, this would help explain the practical ease of simplifying 3-sphere triangulations and would improve known bounds on the number of Pachner moves needed for simplification. The conjecture is supported by the absence of known pathological 3-sphere triangulations requiring more than two extra tetrahedra, but remains open.