Monoids with sub-log-exponential free spectra

A classic result from the 1960s states that the asymptotic growth of the free spectrum of a finite group image is sub-log-exponential if and only if image is nilpotent. Thus a monoid image is sub-log-exponential implies image, the pseudovariety of semigroups with nilpotent subgroups. Unfortunately, little more is known about the boundary between the sub-log-exponential and log-exponential monoids. The pseudovariety image consists of those finite semigroups satisfying (xωyω)ω(yωxω)ω(xωyω)ω≈(xωyω)ω. Here it is shown that a monoid image is sub-log-exponential implies image. A quick application: a regular sub-log-exponential monoid is orthodox. It is conjectured that a finite monoid image is sub-log-exponential if and only if it is image, the finite monoids in image having nilpotent subgroups. The forward direction of the conjecture is proved; moreover, the conjecture is proved for image when image is completely (0)-simple. In particular, the six-element Brandt monoid image (the Perkins semigroup) is sub-log-exponential.