Let $X$ be the universal cover of a closed negatively curved Riemannian manifold. Let $x_0\in X$ be a base point, $S$ be the unit sphere in $T_{x_0}X$ and $\exp:T_{x_0}X\rightarrow X$ be the Riemannian exponential map.

For any measurable subset $C\subset S$ and any $R\in\mathbb{R}_+$, we put $$ [0,R]\cdot C=\{tv\in T_{x_0}X\mid t\in[0,R],\,v\in C\} $$ and define the volume entropy of $C$ as $$ h(C):=\lim_{R\rightarrow+\infty}\frac{1}{R}\log \mathrm{vol}\big(\exp\big([0,R]\cdot C\big)\big). $$ So $h(S)$ is the usual volume entropy of $X$.

I want to know whether the following proposition on measure continuity of $h(C)$ is true.

Proposition. For any $\epsilon>0$, there exists $\delta>0$, such that for any measure subset $C$ of $S$ whose complement has measure less than $\delta$, we have $h(C)>h(S)-\epsilon$.