Completeness of the intermediate-growth scale for finitely generated Lie PI-algebras

Let $L$ be a finitely generated Lie PI-algebra. Write $\gamma_L(n)$ for its growth function, and let $\ln^{(q)}n$ denote the $q$-fold iterated natural logarithm. Intermediate-growth scale conjecture. There exist numbers $q,N_0$ such that

$$\gamma_L(n)\le \exp\bigg( \frac{n}{ \ln^{(q)}n } \bigg),\quad n\ge N_0.$$

The conjecture asserts that the scale governing intermediate growth of finitely generated Lie PI-algebras is complete; the paper establishes completeness of the analogous scale for superexponential codimension growth, while this assertion remains unresolved.