Linear independence of 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 zigzag word is a word in the generators of $T$ whose successive starred and nonstarred elements alternate. Linear-independence conjecture. For every integer $n\geq 1$, each of the following sets is linearly independent: the zigzag words of length $n$ in $T$ that begin with a nonstarred element, and the zigzag words of length $n$ in $T$ that begin with a starred element. This would clarify the structure and linear independence of the natural spanning words in $T$, but the source gives no resolution of the conjecture.