Duality map for biperiodic Fibonacci words

Let $\mathfrak{F}^{(a,b)}$ be the infinite binary biperiodic Fibonacci word with parameters $a,b\geqslant 1$. There is an explicit, effectively computable map $\theta$ on infinite binary words, built from a letter relabelling and a bounded prefix correction. Duality-map conjecture. For all $a,b\geqslant 1$,

$$\mathfrak{F}^{(b,a)}=\theta\bigl(\mathfrak{F}^{(a,b)}\bigr).$$

This would give a uniform explanation of the transformation of combinatorial invariants under exchanging the two parameters, rather than requiring separate arguments for each invariant. The supplied text does not state whether this conjecture has been resolved.