Infinite representations conjecture for automatic presentations of the nonnegative rationals

Let $k\geq 2$ and let $L$ be a regular language of base-$k$ representations of pairs whose quotient set is

$${\rm quo}_k(L)=\mathbb{Q}^{\geq 0}.$$

A representation in $L$ is a word encoding a pair of nonnegative integers and hence a rational number. Infinite representations conjecture. If $L$ is a regular language with ${\rm quo}_k(L)=\mathbb{Q}^{\geq 0}$, then $L$ contains infinitely many distinct representations for infinitely many distinct rational numbers. This concerns the unavoidable multiplicity of representations in any regular presentation of all nonnegative rational numbers; the source states it as a conjecture without resolving it.