Asymptotic formula for the Dirac over-threshold

For integers $k\geq2$ and $m$, let $\bar\eta_{k,m}$ denote the $(k,m)$-Dirac over-exponent, and let $f(\ell_{k,m})$ be the quantity defined in the paper. The asymptotic over-threshold conjecture. There exists a constant $m_k$ such that, for every $m\geq m_k$,

$$\bar\eta_{k,m}=\frac{1}{f(\ell_{k,m})}.$$

The conjecture asserts that the lower bound obtained in the paper gives the exact over-threshold for all sufficiently large path powers. The source presents it as open and notes that an appropriately strengthened version of a preceding lemma would imply it.