  We show that the measure of maximal entropy for the hereditary closure of a
$\mathscr{B}$-free subshift has the Gibbs property if and only if the Mirsky
measure of the subshift is purely atomic. This answers an open question asked
by Peckner. Moreover, we show that $\mathscr{B}$ is taut whenever the
corresponding Mirsky measure $\nu_\eta$ has full support. This is the converse
theorem to a recent result of Keller.
