Conjecture on positivity of the Fibonacci characteristic function

Let $C_a(X)$ denote the characteristic function associated with the first letter $a$ of the Fibonacci substitution. The quantities $_n$ measure the left endpoints of intervals on which the auxiliary characteristic polynomials are positive; the preceding proposition shows that if $_n=-1$ for all sufficiently large $n$, then $C_a(X)>0$ on $(-1,1)$. Positivity conjecture.

$$C_a(X)>0$$

for every $X\in(-1,1)$. This conjecture asserts the first alternative suggested by the preceding analysis and would rule out zeros of $C_a$ throughout the relevant interval.