The Dao-number regularity conjecture for ideals

Let $(R,\mathfrak{m})$ be the local ring under consideration, let $I\subset R$ be an ideal, let $\mathcal{R}(\mathfrak{m})=\bigoplus_{k\ge0}\mathfrak{m}^k$ be the Rees algebra of $\mathfrak{m}$, and let $\mathcal{R}(\mathfrak{m},I)=\bigoplus_{k\ge0}I\mathfrak{m}^k$ be the extension of $I$ in $\mathcal{R}(\mathfrak{m})$. Write $\mathfrak{d}_1(I)$, $\mathfrak{d}_2(I)$, and $\mathfrak{d}_3(I)$ for the Dao numbers of $I$. Dao-number regularity conjecture. For all ideals $I\subset R$:

$$\mathfrak{d}_2(I)\le\mathfrak{d}_3(I)=\mathfrak{d}_1(I)\le\operatorname{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{R}(\mathfrak{m},I).$$

The paper proves this bound under additional hypotheses, namely when $\operatorname{depth}\operatorname{gr}_{\mathfrak{m}}(I)>0$ or when $R$ is regular; the conjecture asserts that the same inequality holds for every ideal in the setting considered.