Consecutive interlacing of noncrossing Chow polynomials

Determine whether the Chow polynomials of the noncrossing partition lattices NC_{n+1} and NC_{n+2} interlace for every n.

Background

The paper proves that each individual noncrossing Chow polynomial H_{NC_{n+1}}(t) is real-rooted by transferring tieless parking-function descent enumerators to finite-alphabet Smirnov words and applying a last-letter interlacing recurrence.

The authors note that this argument operates at fixed alphabet size and arbitrary word length, whereas the noncrossing Chow family follows the diagonal (r,m)=(n,n+1). The unresolved question asks whether real-rootedness can be strengthened to interlacing between consecutive members of this diagonal family. The paper explains that the existing last-letter recurrence does not directly compare consecutive values of n.‌