Asymptotic strictness conjecture for the type of numerical monoids

Let $\mathscr{T}(e,m)$ and $\mathscr{D}(e,m)$ denote the functions defined in the paper for the type and the relevant upper bound of numerical monoids. Asymptotic strictness conjecture. For any fixed value of $e\geq 3$, one has

$$\mathscr{T}(e,m)<\mathscr{D}(e,m)$$

for $m\gg0$. The conjecture proposes that the upper bound for the type is eventually not attained, in analogy with the corresponding asymptotic result for the related function; the preceding corollaries establish equality in several other regimes, but the stated asymptotic strict inequality remains open.