A conjecture about multiple $t$-values

Abstract

For positive integers $a_1,\dots ,a_r$ with $a_1\ge 2$, the multiple $t$-value $t(a_1,\dots ,a_r)$ is defined by the series $\sum _{\genfrac{}{}{0ex}{}{n_1>\dots >n_r>0}{n_i\text{ odd}}}{n}_1^{-a_1}\cdots n_r^{-a_r}$. For an integer $k\ge 2$, the dimension of the $\mathbb{Q}$-vector space generated by all the multiple $t$-values of weight $k$ has been predicted by Hoffman to be the $k$-th Fibonacci number. In this short note we give a conjectural basis of this vector space.

Conjecture is made by considering multiple $t$-values of small weights. Comments are welcome