Reversal invariance conjecture for the Thue–Morse sequence in base $3/2$

Let $x_{3/2}$ be the binary Thue–Morse sequence in base $3/2$. For a finite word $w=w_1\dots w_m$, write $\overleftarrow{w}=w_m\dots w_1$ for its reversal. Reversal invariance conjecture. If the word $w$ occurs in $x_{3/2}$, then $\overleftarrow{w}$ also occurs in $x_{3/2}$. This is one of the conjectured structural symmetries of the sequence; the source provides no resolution.