Positive asymptotic fraction below the b-loading threshold

Let $\mathcal P(n)$ be the set of partitions of $n$, let $\mathbf t\in\mathcal P(n)^3$, and let $g(\mathbf t)$ denote the corresponding Kronecker coefficient. Define

$$b_{\star}=\min\{b(\mathbf t):g(\mathbf t)=0,\ \mathbf t\in\mathcal P(n)^3\}.$$

Let $F_n$ be the fraction of triples $\mathbf t\in\mathcal P(n)^3$ satisfying $b(\mathbf t)<b_{\star}$. Positive-threshold fraction conjecture. The fraction $F_n$ has a nonzero infimum limit as $n\rightarrow\infty$. Since $b(\mathbf t)<b_{\star}$ implies $g(\mathbf t)\neq0$, this would show that a positive asymptotic fraction of triples is certified to have nonzero Kronecker coefficient by the $b$-loading threshold. The claim is supported by numerical data, including approximately $31.8\%$ of triples for $n=20$, but no asymptotic proof is provided.