Hereditary subshifts whose measure of maximal entropy has no Gibbs property

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.