Balanced triangulations attaining the four-dimensional lower bound

Determine whether every PL 4-manifold admits a balanced triangulation in the class \(\mathcal D\) for which each quantity \(f_{\{\varepsilon_i,\varepsilon_{i+2}\}}-f_{\{\varepsilon_i\}}-f_{\{\varepsilon_{i+2}\}}\) equals \(m(\Delta)+2\), thereby attaining the prescribed balanced-genus lower bound.

Background

For a fixed cyclic permutation $$\varepsilon=(\varepsilon_0,\varepsilon_1,\varepsilon_2,\varepsilon_3,\varepsilon_4)$$, the authors define a class $\mathcal D$ of balanced 4-manifolds whose relevant flag face-number expressions attain the lower-bound value $m(\Delta)+2$. A PL 4-manifold with a balanced triangulation in this class achieves the balanced-genus lower bound established in the paper.

The authors emphasize that balanced triangulations of PL 4-manifolds are poorly understood and explicitly conjecture that every PL 4-manifold has such a triangulation.