Bean's informal high-dimensional asymptotics for the full conformal LASSO

Let $(X_i,Y_i)_{i=1}^{n+1}$ follow the data-generating process described in Section 2, let $\hat{\beta}$ be the LASSO estimator fit on all $n+1$ observations, and let $\hat{\beta}_{(n+1)}$ be the corresponding estimator fit on the first $n$ observations. Let $\lambda_i$ and $\epsilon_i$ denote the scale and noise variables, let $\beta_i^*$ denote the population regression coefficients, and suppose that $d/n\to\gamma$. Bean's informal asymptotic conjecture. Under these assumptions, there exist asymptotic constants $N_{\infty}$ and $c_{\infty}$ such that

$$Y_i-X_i^\top\hat{\beta}-\frac{1}{1+2\lambda_{n+1}^2c_{\infty}}(Y_{n+1}-X_{n+1}^\top\hat{\beta}_{(n+1)})\stackrel{\mathbb{P}}{\to}0,$$

and

$$\|\hat{\beta}-\hat{\beta}_{(n+1)}\|_2\stackrel{\mathbb{P}}{\to}N_{\infty}.$$

Moreover, $N_{\infty}$ and $c_{\infty}$ can be computed exactly as solutions of a system of equations depending only on the distributions of $\epsilon_i$, $\lambda_i$, and $\beta_i^*$, the regularization level $\tau$, and the ratio $\gamma$. These asymptotics are heuristic and supported by simulations rather than proved in the paper; they are intended to predict the high-dimensional behaviour of the fitted residuals and coefficients underlying full conformal LASSO inference.