Positive limiting proportion of triples detected by b-loadings

Prove that the fraction of triples \(\mathbf t\in\mathcal P(n)^3\) satisfying \(b(\mathbf t)<b_\star\), where \(b_\star=\min\{b(\mathbf t):g(\mathbf t)=0\}\), has a strictly positive lower limit as \(n\to\infty\).

Background

The paper defines b⋆b_\star as the smallest bb-loading among triples with vanishing Kronecker coefficient. Consequently, any triple with b(t)<b⋆b(\mathbf t)<b_\star must have a nonzero Kronecker coefficient.

For n=20n=20, approximately 31.8% of all triples satisfy this inequality. The authors conjecture that the proportion remains bounded away from zero in the large-nn limit.

References

Although the picture is not totally clear, it supports \begin{conjecture} The fraction of triples satisfying $b(\mathbf t) < b_\star$ has a nonzero infimum limit as $n\rightarrow\infty$. \end{conjecture}

— Mathematical Data Science  (2502.08620 - Douglas et al., 12 Feb 2025) in Section 3, subsection “Distributions conditioned on \(g(\mathbf t)\neq 0\) or \(g(\mathbf t)=0\)”