The product-chain cutoff conjecture for ordered labeled urn chains

Let $\mathbf{X}^{\operatorname{ord,lab}}$ be the labeled ordered chain with parameters $(n,d,\mu)$, with $dn$ labeled balls and a collection $V$ of all slots. Let $\mathbf{X}^{\operatorname{ord,lab}}_{\operatorname{prod}}$ be the product chain on $V^{dn}$ in which each ball moves independently at rate $1/n$, as does a single ball in $\mathbf{X}^{\operatorname{ord,lab}}$.

Ordered labeled product-chain conjecture. The chains $\mathbf{X}^{\operatorname{ord,lab}}$ and $\mathbf{X}^{\operatorname{ord,lab}}_{\operatorname{prod}}$ exhibit cutoff around the same time.

This conjecture is motivated by the analogy with the interchange process and independent simple random walks. The corresponding assertion for the labeled, unordered chain follows from the paper's results, but the ordered labeled assertion remains open.