The finite multiple T-value dimension recurrence conjecture

Let $\mathsf{FMT}_w$ be the $\mathbb{Q}$-vector space generated by finite multiple $T$-values of weight $w$. Finite multiple $T$-value dimension conjecture. For all $k\geq1$,

$$\dim_{\mathbb Q}\mathsf{FMT}_{2k+1}=\dim_{\mathbb Q}\mathsf{FMT}_{2k}+\dim_{\mathbb Q}\mathsf{FMT}_{2k-1}.$$

This is the finite analogue of the recurrence conjectured by Kaneko and Tsumura for ordinary multiple $T$-values. It is based on numerical computation, and no general proof is supplied.