Onset of representation stability for braid-matroid holonomy Lie algebras

Let $A(n)$ be either $cmathrm{OS}(cmathrm{Br}_n)$ or $cmathrm{VG}(cmathrm{Br}_n)$, and define the graded Lie algebra $cmathcal{L}(n)$ by

$$A(n)^!=cmathcal{U}(cmathcal{L}(n)).$$

The graded piece $cmathcal{L}(n)_i$ carries an $cmathfrak{S}_n$-representation whose stability onset is measured in $n$.

Braid-matroid primitive onset conjecture. The onset of representation stability is $n=2i$ for fixed $i eggeq 1$ when $A(n)=cmathrm{OS}(cmathrm{Br}_n)$, and is $n=2i$ for fixed $i eggeq 3$ when $A(n)=cmathrm{VG}(cmathrm{Br}_n)$.

The general representation stability of these Lie algebra families follows from the paper's results, while the stated sharp onset is supported by computation and remains conjectural.