The even all-ones S-value basis conjecture

Let $S_{\mathcal A}$ and $S_\shuffle^{\mathcal S}$ denote the finite and shuffle-regularized symmetric $S$-values, respectively. Even all-ones S-value conjecture. For every even $w\in\mathbb{N}$, there are rational numbers $c_j\in\mathbb{Q}$, $1\leq j\leq w/2$, such that

$$S_{\mathcal A}(\{1\}^w)=\sum_{j=1}^{w/2}c_jS_{\mathcal A}(j,w-j), \qquad S_\shuffle^{\mathcal S}(\{1\}^w)=\sum_{j=1}^{w/2}c_jS_\shuffle^{\mathcal S}(j,w-j).$$

Moreover, the values $S_{\mathcal A}(j,w-j)$ and $S_\shuffle^{\mathcal S}(j,w-j)$ for $1\leq j\leq w/2$ are each $\mathbb{Q}$-linearly independent. This is presented as a further conjecture after the odd-weight formulas, with no resolution supplied.