Accumulation-point conjecture for quotient values of binary sequences

Let ${\bf u}=(u_n)_{n\in\mathbb{N}}$ be a binary sequence that is not eventually periodic. Suppose that the set of return lengths $\mathcal{R}(\mathbf{u})$ is finite, and that

$$\{n\in\mathbb{N}: u_n=0,\ u_{n+1}=u_{n+2}=u_{n+3}=1\}$$

is infinite. Let $f$ be the sequence defined by the paper's equation for $f(n)$, and let $\mathcal{V}(f)$ denote its set of quotient values. Accumulation-point conjecture. For almost all pairs $(a,b)$ in the sense of asymptotic density, the set $\mathcal{V}(f)$ has infinitely many accumulation points. More precisely, its set of accumulation points in $\mathbb{R}\cup\{\infty\}$ is homeomorphic either to a union of the Cantor set and a finite set or to the whole set $\mathbb{R}\cup\{\infty\}$. The conjecture concerns the structure of quotient values generated by non-eventually-periodic binary sequences; the asserted almost-everywhere conclusion and the alternatives for the accumulation set remain unproved in the supplied text.