Boxicity of divisor graphs on more general vertex sets

Determine the boxicity of divisor graphs whose vertex sets are more general subsets of the positive integers, including $(n\alpha,n]$, the arithmetic-progression sets $a[n]+b=\{ak+b:k\in[n]\}$, and $(\alpha n,n]$ for 0<\alpha<1.

Background

The paper studies the special divisor graph D(n), whose vertices are all positive divisors of a fixed natural number n. It proposes extending this analysis to divisor graphs induced by broader classes of integer sets.

The listed sets are singled out because Lewis et al. previously investigated the poset dimension of the corresponding divisibility posets. The unresolved issue is to determine the corresponding boxicity for these more general divisor-graph constructions.