Arnold's average-density conjecture for numerical semigroups

Let ${\bf d}^m=(d_1,\ldots,d_m)$ be a vector of generators, and let ${\sf S}({\bf d}^m)$ be the associated numerical semigroup with conductor ${\sf C}({\bf d}^m)$. Let $\widetilde{G}({\bf d}^m)$ denote the number of elements of the semigroup in the segment from $0$ to ${\sf C}({\bf d}^m)-1$, and define

$$p({\bf d}^m)=\frac{\widetilde{G}({\bf d}^m)}{{\sf C}({\bf d}^m)}.$$

Arnold's average-density conjecture. For large ${\bf d}^m$, with overwhelming probability, this fraction is asymptotically $1/m$; equivalently,

$$\widetilde{G}({\bf d}^m)\stackrel{\mathrm{asymptotically\ weak}}{\equiv}\frac{1}{m}{\sf C}({\bf d}^m).$$

This is Arnold's Conjecture #1999–9 concerning the average distribution of a numerical semigroup below its conductor. The supplied text gives no resolution, so the conjecture remains open.