Dao–Celikbas conjecture on the third Dao number and reduction numbers

Let $(R,\mathfrak{m})$ be a Cohen–Macaulay local ring with infinite residue field and dimension at least $2$. A minimal reduction $I$ of $\mathfrak{m}$ is a reduction minimal under inclusion. Dao–Celikbas conjecture. For every minimal reduction $I$ of $\mathfrak{m}$,

$$\mathfrak{d}_3(I)={\rm r}_I(\mathfrak{m}).$$

This extends the corresponding dimension-one result to higher-dimensional Cohen–Macaulay local rings. The source does not provide evidence of a resolution, so the conjecture is recorded as open.