Symmetrical bound for the MDL code in terms of shortest-grammar length

Let $v$ be a string, let $L^m(v)$ denote the length of its shortest grammar, and let $\boldsymbol{\operatorname{MDL}}(v)$ denote the MDL code for $v$. Let $\gamma(n):=n f_1(n)$. Symmetrical-bound conjecture. There is an inequality

$$\left|\operatorname{len}\operatorname{MDL}(v)-\gamma\bigl(L^m(v)\bigr)\right|\leq f_2\bigl(L^m(v)\bigr),$$

where the functions $f_i\geq 0$ satisfy

$$0\leq f_i(n+1)-f_i(n)\leq \frac{c_i}{n}$$

for some constants $c_i$. This proposed bound is intended to relate the MDL code length to the shortest-grammar length and is used in the paper's heuristic argument about vocabulary and excess code length; the source calls it probable rather than proving it.