General-number-of-parts conjecture for discrete stick fragmentation

Fix an integer $k\geq2$. In the discrete stick-fragmentation process, break each stick into $k$ pieces by choosing $k-1$ cut points recursively according to the uniform distribution, allowing fewer than $k$ pieces when the stick becomes too short. Fix $n=tk$ for an integer $t\geq1$ and a subset $S\subset\{0,\dots,n-1\}$ of size $(t-1)k$, and let

$$\mathfrak{S}=\{1\}\cup\{m\in\mathbb{Z}_+:m=qn+r,\ r\in S,\ q\in\mathbb{Z}\}.$$

Starting with $R$ identical sticks of positive integer length $L\notin\mathfrak{S}$, the collection of ending stick lengths converges to strong Benford behavior as $L\to\infty$ whenever $R>f(L)$ for some function $f(L)\to\infty$. General-number-of-parts conjecture. Moreover, if the number of residue classes in the stopping set is not $(t-1)k$, the resulting stick lengths do not converge to strong Benford behavior. The claim is a conjectural extension of the two-piece process, supported by simulations; the appropriate growth condition and the non-Benford assertion remain open.