Fast convergence rate conjecture for guessing numbers

Let G_{[?] be the graph associated with the term-coding system, and let $S_n”$ denote its number of solutions of size $n$. Define

$$L=\lim_{n\to\infty}\mathrm{Guess}(G_{\Gamma”},n).$$

Fast convergence rate conjecture. The convergence satisfies

$$L-\mathrm{Guess}(G_{\Gamma”},n)=L-\log_n(S_n”)=O\left(\frac{1}{\log n}\right).$$

A rate of this form would provide quantitative control of the convergence guaranteed by Fekete's lemma and would help establish the precise asymptotic behaviour $S_n=\Theta(n^L)$. The source presents this as a working hypothesis, and no resolution is supplied.