On the Waldschmidt constant of square-free principal Borel ideals

Fix a square-free monomial $m\in S=\mathbb{K}[x_1,\dots ,x_n]$. The square-free principal Borel ideal generated by $m$, denoted $\mathrm{s}\mathrm{f}\mathrm{B}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{l}(m)$, is the ideal generated by all the square-free monomials that can be obtained via Borel moves from the monomial $m$. We give upper and lower bounds for the Waldschmidt constant of $\mathrm{s}\mathrm{f}\mathrm{B}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{l}(m)$ in terms of the support of $m$, and in some cases, exact values. For any rational $\frac{a}{b}\ge 1$, we show that there exists a square-free principal Borel ideal with Waldschmidt constant equal to $\frac{a}{b}$.