High-dimensional coverage conjecture for full conformal quantile regression

Let $\ell_{\alpha}(u)=\alpha u$ for $u\geq 0$ and $(\alpha-1)u$ for $u<0$ be the pinball loss, let $\hat{\eta}_{n+1}$ be the dual quantile-regression score, and let $R_{n+1}^{(n+1)}=Y_{n+1}-\hat{\beta}_{0,(n+1)}-X_{n+1}^{\top}\hat{\beta}_{(n+1)}$ be the leave-one-out test residual. Let $\lambda$ and $\epsilon$ be a copy of $(\lambda_i,\epsilon_i)$ independent of $Z\sim N(0,1)$, and let $\gamma=\lim d/n$. High-dimensional quantile-regression conjecture. Under the data-generating assumptions of the paper, there exist constants $c_{\infty}$, $N_{\infty}$, and $\beta_{0,\infty}$ such that

$$\hat{\eta}_{n+1}-\frac{R_{n+1}^{(n+1)}-\operatorname{prox}(\lambda_{n+1}^2c_{\infty}\ell_{\alpha})(R_{n+1}^{(n+1}))}{\lambda_{n+1}^2c_{\infty}}\stackrel{\mathbb{P}}{\to}0,$$

and

$$R_{n+1}^{(n+1)}\stackrel{D}{\to}\epsilon-\beta_{0,\infty}+\lambda N_{\infty}Z.$$

Writing $W=\epsilon-\beta_{0,\infty}+\lambda N_{\infty}Z$, the triple satisfies the displayed system of three equations in the source, and the dual full conformal set has asymptotic conditional coverage $1-\alpha$, whereas standard quantile regression has limiting conditional coverage $\mathbb{P}(W\leq 0)$. The conjecture gives a heuristic high-dimensional characterization of quantile-regression conformal coverage; rigorous proofs remain open.