  Which $4$-manifolds admit a flag-no-square (fns) triangulation? We introduce
the "star-connected-sum" operation on such triangulations, which preserves the
fns property, from which we derive new constructions of fns $4$-manifolds. In
particular, we show the following: (i) there exist non-aspherical fns
$4$-manifolds, answering in the negative a question by Przytycki and
Swiatkowski; (ii) for every large enough integer $k$ there exists a fns
$4$-manifold $M_{2k}$ of Euler characteristic $2k$, and further, (iii) $M_{2k}$
admits a super-exponential number (in $k$) of fns triangulations - at least
$2^{\Omega(k \log k)}$ and at most $2^{O(k^{1.5} \log k)}$.
