Suppose that one has the usual regression model $\mathbf{y} = \mathbf{X} \boldsymbol{\beta} + \boldsymbol{\varepsilon}$, where each $\varepsilon_t$ is iid distributed with $\mathbb{E}(\varepsilon_t) = 0$ and $\mathbb{V}\text{ar}(\varepsilon_t) = \sigma^2$. Assume also that variables in $\mathbf{X}$ are non-stochastic for simplicity. It is well-known that asymptotically $$T:=\frac{(n-k)s^2}{\sigma^2} \sim \chi_{n-k},$$ where $s^2 := 1 / (n-k) \sum_t e_t^2$, and $e_t$ denotes least-squares residuals, $n$ is the sample size, and $k$ is the number of columns of $\mathbf{X}$.
Usually when one wants to construct a confidence interval for some $\beta_j$, one uses the so-called studentized bootstrap confidence interbal, which makes use of the $t$-statistic (see, e.g, Davison and Hinkley for details). Is it possible to improve upon the regular percentile bootstrap for the variance $\sigma^2$ by making use of the above statistic $T$? Namely for each $i = 1, \dots, B$, resample the data in some way, compute the statistic $\hat{T}_{(i)} = (n-k) s^2_{(i)}/s^2$, where $s^2$ is the original estimate, order the $\hat{T}_{(i)}$'s, and then construct the interval, as in the standard case.
