Tribonacci dimension conjecture for alternating multiple two-one values

For each weight $n$, let $\mathsf{AMTV}_n$ denote the rational vector space of alternating multiple two-one values of weight $n$. Tribonacci dimension conjecture. The generating function of their dimensions is

$$\displaystyle\!\sum_{n=0}^{\infty}\bigl(\dim_{\mathbb Q}\mathsf{AMTV}_n\bigr)t^n=\displaystyle\!\frac{1}{1-t-t^2-t^3}.$$

Equivalently, the dimensions form the tribonacci sequence $\{d_w\}_{w\ge1}=\{1,2,4,7,13,24,\dots\}$. The paper says this conjecture is supported by theoretical and numerical evidence and treats it as unsolved.