Homogeneous ideals with minimal singularity thresholds

Let $({\mathcal{O}}_n,\mathfrak{m})$ denote the ring of germs of holomorphic functions ${\mathbb{C}}^n\to \mathbb{C}$, and let $I\subseteq {\mathcal{O}}_n$ be an $\mathfrak{m}$-primary ideal. Demailly and Pham showed that $\mathrm{l}\mathrm{c}\mathrm{t}(I)\ge \frac{1}{e_1(I)}+\cdots +\frac{e_{n-1}(I)}{e_n(I)}$, where $e_j(I)$ is the mixed multiplicity $e(I,\dots ,I,\mathfrak{m},\dots ,\mathfrak{m})$, with $I$ repeated $j$ times and $\mathfrak{m}$ repeated $n-j$ times. We generalize the lower bound to the case of an arbitrary ideal of an excellent regular local (or standard-graded) ring of equal characteristic, with $\mathrm{l}\mathrm{c}\mathrm{t}(I)$ replaced by the $F$-threshold $c^{\mathfrak{m}}(I)$ in positive characteristic. Our main result is a classification of homogeneous ideals in polynomial rings for which the lower bound is attained, resolving a conjecture of Bivià-Ausina in the graded case.