Živaljević's conjecture on LC triangulations of simply connected smooth manifolds

A smooth $d$-manifold is a smooth manifold of dimension $d$. A triangulation is locally constructible (LC) if it can be obtained from a tree of $d$-simplices by repeatedly identifying two adjacent $(d-1)$-simplices in the boundary. A manifold is simply connected when its fundamental group is trivial.

Živaljević's conjecture. Every simply connected smooth $d$-manifold admits some LC triangulations.

The source attributes this conjecture to Živaljević and does not state a resolution. It is a restatement of the private-communication conjecture appearing earlier in the paper and is therefore merged here rather than emitted as a separate row.