Akiyama's conjecture on triple expansions in rational bases

Let $p>q$ be coprime integers. An expansion of a positive real number $x$ in rational base $p/q$ is an infinite word satisfying the rational-base language and valuation conditions described in the source. Akiyama's conjecture. Every positive real number $x$ admits at most two expansions in rational base $p/q$. The supplied text notes that this is proved when $p\geq 2q-1$, while the general case remains open.