Hyperbolic continuation of high-entropy homoclinic classes

Let $f$ be a $C^r$ diffeomorphism of a compact surface. For $0<t\le h_{\rm top}(f)$, let $\mathcal H(f,t)$ be the set of homoclinic classes of measures $C$ satisfying $h(f,C)\ge t$. Let $f_n\to f$ converge in the $C^r$ topology. For a homoclinic class $C$ of $f$, its hyperbolic continuation under a nearby diffeomorphism $g$ is the homoclinic class $C_g$ containing all ergodic hyperbolic measures of $g$ homoclinically related to the hyperbolic continuation of a periodic orbit $O$ with $\mu_O\in C$. Continuation conjecture. Fix $t>\lambda_{\min}(f)/r$. If $\#\mathcal H(f_n,t)=\#\mathcal H(f,t)$ for all large $n$, then, for all large $n$, each homoclinic class in $\mathcal H(f_n,t)$ is the hyperbolic continuation of some homoclinic class in $\mathcal H(f,t)$. This conjecture refines upper semicontinuity by asserting that equality of the numbers of high-entropy classes forces every perturbed class to arise by continuation from an original one.