Basis conjecture for nonrepeating zigzag words in the algebra T

Fix a nonnegative integer $d$, a sequence $(\{\theta_i\}_{i=0}^d;\{\theta^*_i\}_{i=0}^d)$ of scalars in $\mathbb{K}$ satisfying the distinctness condition and the stated recurrence condition, and let $T$ be the corresponding $\mathbb{K}$-algebra. A word $x_1x_2\cdots x_n$ in $T$ is nonrepeating when $\overline{x}_{i-1}\ne\overline{x}_i$ for $2\leq i\leq n$; a nontrivial nonrepeating zigzag word is such a word, other than the trivial words excluded by the source's terminology. Basis conjecture. Each of the following is a basis for the $\mathbb{K}$-vector space $T$: the nontrivial nonrepeating zigzag words beginning with a nonstarred element; those beginning with a starred element; those ending with a nonstarred element; and those ending with a starred element. This conjectural basis would give an explicit word basis for $T$ and strengthen the preceding linear-independence assertion; the source gives no resolution of it.