Long-range dependence conjecture for the limiting variance of the tail dependence ratio estimator

Let $(X_t)_{t \in \mathbb{Z}}$ be a max-stable stationary time series with $\alpha$-Fréchet margins and pairwise extremal coefficients $(\theta_t)_{t \in \mathbb{Z}}$. Assume that $2-\theta_t=Ct^{-\delta}$ for $t\in\mathbb{Z}$, where $C>0$ and $\delta\in(0,1)$. Let $\widehat{p}_{n,b_n}(u_n)$ denote the marginal exceedance-probability estimator and $\widehat{\mathbb{P}}_{T,n,b_n}(u_n)$ the corresponding joint exceedance estimator, and consider their ratio estimator $\widehat{\chi}_{T,b_n}$. The limiting-variance conjecture. Under appropriate assumptions on the asymptotic covariance of $\widehat{p}_{n,b_n}(u_n)$ and $\widehat{\mathbb{P}}_{T,n,b_n}(u_n)$, there exists a constant $K>0$ such that, for every real sequence $u_n\to\infty$ and every integer sequence $b_n\to\infty$,

$$\lim_{n\to\infty}\operatorname{Var}\left(\sqrt{n^\delta p(u_n)}\frac{\widehat{\mathbb{P}}_{T,n,b_n}(u_n)}{\widehat{p}_{n,b_n}(u_n)}\right)=K.$$

The conjecture formalizes the expectation that the ratio estimator has the same long-range-dependence variance rate as its denominator; a Delta-method argument might establish it when the covariance and Taylor-expansion remainder conditions are sufficiently strong, but those conditions are not specified here.