Uniform entropy gap for a fixed quotient

Establish whether every fixed quotient K=L(g₀) with h_top(Φ|_K)>h_top(f) admits a uniform positive constant δ(K) such that every C∞ diffeomorphism g isotopic to f with L(g)=K satisfies h_top(g) ≥ h_top(Φ|_K)+δ(K).

Background

The strict entropy-gap theorem gives a positive gap between representative entropy and quotient entropy for each individual C∞ representative whose quotient entropy exceeds the pseudo-Anosov minimum. It does not show that the gap is bounded away from zero uniformly over all smooth representatives realizing the same quotient K. The question asks whether the quotient entropy can be approached by entropies of smooth realizations without being attained.

References

Does there exist \delta(K)>0 such that h_{\mathrm{top}(g)\ge h_{\mathrm{top}(\Phi|_K)+\delta(K) for every C\infty diffeomorphism g\simeq f satisfying L(g)=K?

— Entropy and periodic points in pseudo-Anosov isotopy classes  (2610.26230 - Carrasco et al., 19 Oct 2026) in Section 6, question titled “Entropy gap”

Is this relation dynamically relevant in some sense?

— Entropy and periodic points in pseudo-Anosov isotopy classes  (2610.26230 - Carrasco et al., 19 Oct 2026) in Appendix, Section “Questions {content} Comments”