$\mathbf{F}$-jumping numbers can be irrational

Let $k$ be an $F$-finite and infinite field of characteristic $p>2$. We show, there exist infinitely many $F$-finite local domains $(R,\mathfrak{m})$ which are not $\mathbb{Q}$-Gorenstein and ${\tau }_{\mathrm{b}}(R;{\mathfrak{m}}^t)$ has all but finitely many \emph{irrational} $F$-jumping numbers.