The uniform distribution hypothesis for extremal B-star[g] sets

Let $R(g,n)$ denote the maximum size of a $B^{\ast}[g]$ set contained in $\{1,\dots,n\}$. A sequence $(S_n)_{n=1}^\infty$ of sets of positive integers becomes uniformly distributed if

$$\limsup_{n\to\infty}\sup_{0\leq\alpha<\beta\leq1}\left|\frac{|S_n\cap[\alpha M_n,\beta M_n]|}{|S_n|}-(\beta-\alpha)\right|=0,$$

where $M_n$ is the largest element of $S_n$. Uniform distribution hypothesis. Let $g\geq2$ be an integer. If $S_n\subseteq\{1,\dots,n\}$ is a sequence of $B^{\ast}[g]$ sets with $|S_n|\sim R(g,n)$, then $(S_n)$ becomes uniformly distributed. The assertion generalizes the known result for $g=2$; its resolution for general $g$ is not supplied in the source.