Conjectural basis for box-product spaces of qMZV words

For $1\leq z\leq d$, let $\mathfrak{M}_{z,d}$ be the recursively defined set of plain basis vectors, and define

$$\mathfrak{B}_{z,d}:=\left\{u_{n_1}\cdots u_{n_s}\boxast u_{\ell_1}\cdots u_{\ell_d}:\begin{array}{l}u_{\ell_1}\cdots u_{\ell_d}\in\mathfrak{M}_{z,d},\ n_j\in\mathbb{Z}_{>0},\\ n_1+\cdots+n_s+\ell_1+\cdots+\ell_d=z+d,\\ \ell_1>z-d+\delta_{n_1=s=1}\end{array}\right\}.$$

Basis conjecture. For $1\leq z\leq d$, $\mathfrak{B}_{z,d}$ is a basis of $\mathscr{S}_{z,d}$. The set was obtained from numerical calculations, and the paper postpones a proof to future work.