The density conjecture for the sets $S_{b,m}$ in SALIQUANT

For $b>0$, let $S_{b,m}$ denote the set of inputs associated with the nim-value $f(a,b,m)$ in SALIQUANT. The density conjecture for $S_{b,m}$. If $m=0$, then $S_{b,m}$ has positive density; if $m>0$, then $S_{b,m}$ has density $0$ but is nonempty. This conjecture refines the upper bound that the density of $S_{b,m}$ is at most $\frac{1}{2m+1}$, and predicts that the experimentally observed nonzero values with $m>0$ are sparse while the value $m=0$ occurs with positive density.