The author extends to the boundary case combinatorial relations first introduced and studied in a series of papers [the reviewer, Proc. Am. Math. Soc. 105, 1008-1014 (1989; Zbl 0683.57007); Trans. Am. Math. Soc. 331, 203-214 (1992; Zbl 0762.57014); and with M. Meschiari, Cah. Topologie Géom. Différ. Catégoriques 34, 37-56 (1993; Zbl 0777.57009)] for closed PL 4-manifolds. As a consequence, the author proves that \(\mathbb{D}^2\)-bundles over \(\mathbb{S}^2\) constitute an infinite family of bounded PL 4-manifolds having the same regular genus three.