Conjecture on the block max-min estimator's pivotal limit distribution

Let $f_0$ be the monotone regression function, let $x_0$ be the point of inference, and let $\widehat{f}_n^{-}(x_0)$ be the block max-min estimator with associated block count $n_{\widehat{u},\widehat{v}}^{-}(x_0)$. Let $\sigma$ denote the noise standard deviation, and let $\bm{\alpha}$ and $K(f_0,x_0)$ be as in Theorem. Under the same settings as that theorem, there is a finite random variable $\mathbb{L}_{\bm{\alpha}}^{-}$ that does not depend on $K(f_0,x_0)$. Block max-min pivotal-limit conjecture.

$$\sqrt{n_{\widehat{u},\widehat{v}}^{-}(x_0)}\big(\widehat{f}_n^{-}(x_0)-f_0(x_0)\big)\rightsquigarrow \sigma\cdot \mathbb{L}_{\bm{\alpha}}^{-}.$$

If true, this would provide an asymptotic pivotal quantity for confidence intervals based on the block max-min estimator alone, potentially reducing the computational cost relative to the block average procedure. The source gives no resolution of this conjecture.