The mixed-alphabet language conjecture for smooth words

Let $\mathpzc{A}=\{a,b\}$ be a mixed binary alphabet, let $\mathcal{C}^\infty$ be the set of infinite smooth words over $\mathpzc{A}$, let $\mathcal{C}_f^\infty$ be the set of finite f-smooth words, and let $\mathscr{L}(x)$ and $p_x(n)$ denote the factor language and factor complexity of $x$. Mixed-alphabet language conjecture. Over mixed alphabets, every smooth word $x$ satisfies

$$\mathscr{L}(x)=\mathcal{C}_f^\infty,$$

and, in particular,

$$p_x(n)=p_{\mathcal{C}^\infty}(n)=p_{\mathcal{C}_f^\infty}(n).$$

The conjecture is linked to recurrence and would imply complement and reversal invariance of every smooth-word language; the source contrasts it with failures over even and odd alphabets.