LC triangulation conjecture for Mazur's 4-manifold

Let $W$ denote Mazur's simply connected smooth $4$-manifold with boundary. An LC triangulation is a triangulation obtained by the local constructibility procedure described in the paper.

Mazur-manifold LC triangulation conjecture. Mazur's $4$-manifold admits an LC triangulation.

This is proposed as a first step toward extending Živaljević's conjecture to simply connected $4$-manifolds with boundary. The source leaves the statement open and notes that Mazur's manifold is simply connected but not geometrically $1$-connected.