Asymptotic constant conjecture for t-dimension of interval divisibility posets

For $0<\alpha<1$, let $\mathcal{D}_{(\alpha n,n]}$ be the divisibility poset on the integers in $(\alpha n,n]$. For an integer $t\geq2$, let $\dim_t(P)$ denote the $t$-dimension of a poset $P$. Asymptotic constant conjecture. For each $0<\alpha<1$ and $t\geq2$, there exists a constant $c=c(\alpha,t)$ such that

$$\dim_t\big(\mathcal{D}_{(\alpha n,n]}\big)\sim c\log n$$

as $n\to\infty$. The paper has already established bounds of order $\log n$ for fixed $\alpha$ and $t$, but does not determine whether the ratio to $\log n$ converges to a constant; that convergence is the conjectural part.