Derivation conjecture for the algebra of multiple t-values

Let $\operatorname{QSym}$ be the algebra of quasisymmetric functions, let $A_-:\operatorname{QSym}\to\operatorname{QSym}$ be the derivation defined by $A_-(1)=0$ and

$$A_-(M_{(i_1,\dots,i_k)})=\begin{cases} M_{(i_1,\dots,i_{k-1})},&\text{if }i_k=1,\\ 0,&\text{otherwise},\end{cases}$$

let $\mathfrak H^0$ be the relevant subalgebra, and let $\theta$ map $\mathfrak H^0$ to the $\mathbb Q$-subalgebra $\mathcal T\subset\mathbb R$ generated by $1$ and the multiple $t$-values. Derivation conjecture. The algebra $\mathcal T$ admits a derivation $d$ such that

$$d\theta=\theta A_-.$$

In terms of the tabulated formulas, $d$ corresponds to formal differentiation with respect to $\log 2$. The conjecture proposes that the derivation on quasisymmetric functions descends through $\theta$ to the algebra of multiple $t$-values; the analogous derivation cannot exist for the algebra generated by multiple zeta values.