Spectral condition for ASEP

Establish whether the spectral condition fref{cond} is satisfied for the asymmetric exclusion process with $k$ particles on a segment of size $n$, in the regime $k/n\to\alpha$.

Background

The asymmetric exclusion process considered in the paper consists of particles on a finite segment that attempt right and left unit steps with probabilities $p>1/2$ and $q=1-p<1/2$, subject to the exclusion rule. The paper cites cutoff and limit-profile formulas in the regime where the particle density converges to $\alpha$.

The limit profile is known and is continuous, but the paper explicitly states that the spectral f2 analysis around cutoff remains unresolved. Consequently, whether the eigenvalue-based condition \eqref{cond} is satisfied is unknown.