Asymptotic optimizer sequence for the lower bound in $g(n,k,\ell)$

For each triple $(n,k,\ell)$, let the right-hand side of be the lower-bound expression from Theorem, optimized over the parameter $j$. Asymptotic optimizer conjecture. For every $\ell \geq 2$, there is a sequence

$$0 = \alpha_0(\ell) < \alpha_1(\ell) < \alpha_2(\ell) < \dots < \frac{1}{3}$$

of real numbers such that, if $n,k$ tend to infinity with $\frac{k}{n} \rightarrow \gamma$ and $\alpha_{i-1}(\ell) < \gamma < \alpha_i(\ell)$, then the right-hand side of attains its maximum at $j=i$.

This predicts a phase diagram for the asymptotically optimal choice of $j$ in the lower-bound construction. The source gives no resolution or supporting result for this assertion.