The odd-height all-ones multiple T-value conjecture

Let $T_{\mathcal A}$ and $T_\shuffle^{\mathcal S}$ denote the finite and shuffle-regularized symmetric multiple $T$-values, respectively; let $S_{\mathcal A}$ and $S_\shuffle^{\mathcal S}$ denote the corresponding $S$-values, and let $\beta_w$ be the finite odd-weight beta value. Odd all-ones multiple $T$-value conjecture. For every odd $w\in\mathbb{N}$,

$$T_{\mathcal A}(\{1\}^w)=-S_{\mathcal A}(\{1\}^w)=\frac{2^{w-1}-1}{2^{2w-2}}\beta_w, \qquad T_\shuffle^{\mathcal S}(\{1\}^w)=-S_\shuffle^{\mathcal S}(\{1\}^w)=\frac{2^{w-1}-1}{2^{2w-2}}\zeta(w).$$

The relation is proved for $w=3,5,7$ in the source, but remains conjectural for general odd $w$.