Binomial approximation conjecture for the Poisson bootstrap quantile index distribution

Let the sample $\mathbf{y}$ consist of elements that can be uniquely ordered, and let the quantile $q\in(0,1)$ be fixed. Let $\psi$ denote the index of the original order statistic observed as the sample quantile in the Poisson bootstrap sample. Binomial approximation conjecture. The distribution of $\psi$ can be approximated by a $\operatorname{Bin}(N+1,q)$ distribution in the sense that

$$\max_i\left|p(\psi=i)-p(X=i)\right|\longrightarrow 0 \quad\text{as}\quad N\longrightarrow\infty,$$

where $X\sim\operatorname{Bin}(N+1,q)$. This approximation is proposed because it appears to fit the index distribution well, improves with increasing $N$, and enables fast, resampling-free bootstrap inference for quantiles; the source demonstrates its merit through Monte Carlo simulations, but does not provide a proof or resolution of the conjecture.