Zooming-in conjecture at the maximal distance from the origin for Lévy processes

Assume that the Lévy process $X$ has finite lifetime $\zeta\in(0,\infty)$. Let

$$\tau:=\sup\left\{t\geq 0:\max\left(\|X_t\|,\|X_{t-}\|\right)=\sup_{s\geq 0}\|X_s\|\right\}$$

be the last time at which the Euclidean norm is maximal. Set $M:=X_\tau$ if $\|X_\tau\|\geq\|X_{\tau-}\|$ and $M:=X_{\tau-}$ otherwise, and define the reversed and forward processes by

$$\overleftarrow{X}_t:=\begin{cases}X_{(\tau-t)-}-M,&0\leq t<\tau,\dagger,&t\geq\tau,\end{cases}\qquad \overrightarrow{X}_t:=\begin{cases}X_{\tau+t}-M,&0\leq t<\zeta-\tau,\dagger,&t\geq\zeta-\tau.\end{cases}$$

Let $B$ be the Brownian part of $X$ with a non-singular covariance matrix. Zooming-in conjecture at the maximal distance.

$$\sqrt{n}\bigl(\overleftarrow{X}_{\cdot/n},\overrightarrow{X}_{\cdot/n}\bigr)\xrightarrow{d}(-B^\Downarrow,B^\Uparrow),$$

where the limit pair is a mixture of $(-B^{\downarrow},B^{\uparrow})$ for the independent direction $\eta=-M$. This conjecture predicts a multidimensional analogue of the one-dimensional stable-convergence result for zooming in at an extremal time; proving it is described as exceedingly challenging, and the anticipated stable convergence is not included in the statement.