Invariance of Lyapunov exponents under limits of tidy perturbations

Let $f$ be the diffeomorphism under consideration, and let $g_i=h_i f h_i^{-1}$ be tidy perturbations of $f$ converging to $g=hfh^{-1}$, where the diffeomorphisms $h_i$ converge to $h$. For an ergodic $f$-invariant measure $bc$, write $h_*(bc)$ for its pushforward under $h$, and consider the Lyapunov exponents of these measures for the corresponding diffeomorphisms. Lyapunov-exponent invariance conjecture. Given any ergodic invariant measure $bc$ of $f$, the Lyapunov exponents of $h_*(bc)$ for $g$ are the same as those of $bc$ for $f$. This conjecture concerns the limitation of successive tidy perturbations: although they can alter the dynamics while preserving the relevant orbit structure, the authors expect that such limits cannot change Lyapunov exponents. Whether this invariance holds in the stated generality is left open.