  Given a monomial ideal $I$ with minimal free resolution $\mathcal{F}$
supported in characteristic $p>0$ on a CW-complex $X$ with regular
$2$-skeleton, we construct a CW-complex $Y$ that also supports~$\mathcal{F}$
and such that the face poset $P(Y)$ also supports $\mathcal{F}$ in the sense of
Clark and Tchernev.
