All non-periodic points satisfy the Poisson limit law under exponential mixing

Let $f$ be an exponentially mixing dynamical system, and let $x$ be a non-periodic point. Let $y$ be distributed according to the invariant measure $\mu$, and let $d_n^{(r)}(x,y)$ denote the distance from $x$ to the $r$-th closest point among the orbit segment $\{f^k(y):k\in[1,n]\}$. Proposition (a) asserts that, for almost all $x$, the number of visits of $\{f^k(y)\}_{k\in[1,\tau\rho^{-d}]}$ to $B(x,\rho)$ converges to a Poisson distribution with parameter $\tau\gamma(x)$ as $\rho\to0$, and that the rescaled order statistics form a Poisson process.

Exponential-mixing conjecture. If $f$ is exponentially mixing, then the conclusion of Proposition (a) holds for all non-periodic points $x$.

This would strengthen the almost-everywhere conclusion of Proposition (a) by identifying every non-periodic point as a point where the Poisson limit law holds. The supplied text gives no resolution of this question.