Pure rough path signature-tail asymptotics conjecture

Let $(V,\|\cdot\|)$ be a finite-dimensional Banach space, and let the tensor products be equipped with given reasonable tensor algebra norms. Let $l$ be a Lie polynomial of degree $m$, and write $\pi_n$ for projection to $V^{\otimes n}$. For a pure $m$-rough path ${\bf X}_t=\exp(tl)\in G^{(m)}(V)$, the signature tail asymptotics $L_m({\bf X})$ is defined using the degree-$n$ signature components; in the algebraic formulation this quantity is

$$\limsup_{n\rightarrow\infty}\left(\left(\frac{n}{m}\right)!\|\pi_n(\exp(l))\|\right)^{\frac{m}{n}}.$$

Pure rough path signature-tail asymptotics conjecture. For every pure $m$-rough path, the tail asymptotics equals its local $m$-variation, namely

$$L_m({\bf X})=\|\pi_m(l)\|.$$

This extends the bounded-variation formula and is consistent with the Brownian-motion case. The result is known for $m=1$, and for $V=\mathbb{R}^2$ with the $l^1$-norm it is established for $m=2,3$ and some cases in degrees $4,5$; the general statement remains open.