A faster zigzag sampling algorithm for strongly log-concave distributions

Let $U$ satisfy Assumption, let $\mu$ be the target distribution, let $\rho(X)$ denote the law of an algorithm's output $X$, let $\mu_0$ be the initial distribution, let $\kappa$ be the relevant condition parameter, and let $\varepsilon>0$. Assume that $\kappa$ and $\log(1/\varepsilon)$ are both smaller than some algebraic power of $d$. Zigzag sampling complexity conjecture. There exists an algorithm producing a random variable $X$ such that

$$\chi^2(\rho(X) \,\Vert\, \mu)\leq \varepsilon.$$

Moreover, with high probability, the algorithm requires

$$O\Bigl(d\kappa\log d\bigl(\log\frac{1}{\varepsilon}+\log\chi^2(\mu_0\,\Vert\,\mu)\bigr)\Bigr)$$

evaluations of partial derivatives of $U$. This would improve the computational complexity of the proposed zigzag sampling method by avoiding the extra $O(\sqrt{d})$ factor in the bouncing-rate control; the conjecture remains unproved in the source.