The determinant-scaling conjecture for Stratonovich signatures of linear Brownian motions

Let $(B_t)_{t\geq 0} = (B_t^1, \dots, B_t^d)_{t\geq 0}$ be a $d$-dimensional Brownian motion and let $V$ be an invertible $d\times d$ matrix. For $X=VB$, write $L^{\mathbb{X},\operatorname{Strat}}_{s,t}$ for the relevant Stratonovich signature tail quantity. Determinant-scaling conjecture. One has

$$L^{\mathbb{X},\operatorname{Strat}}_{s,t}=|\det V|\kappa_d(t-s),$$

where $\kappa_d$ depends only on the dimension $d$ and the choice of tensor norms. The conjecture is motivated by simulations for two-dimensional Brownian motion with differently scaled coordinates; proving sharp lower bounds and extending the result beyond this setting remain open problems.