The doubled-prefix conjecture for reduced banned words

Let $1/2<\rho<1$ be a scaling factor, and let $\{\alpha^i\}_{i\in I}$ be the reduced banned words, enumerated in nondecreasing order of length, with $I\subseteq\mathbb{N}$. For each $i\in I$, write

$$\gamma^i=\alpha^i|_{[1,|\alpha^i|-1]}.$$

A doubled-prefix conjecture asserts that, for every $i\in I$, the concatenation $\gamma^i\gamma^i$ does not contain any $\alpha^j$ with $j\leq i$ as a subword.

This condition is intended to support the observed recursive pattern in the reduced banned words and the converse of the sufficient condition for an iterated function system to be of finite type. The source says that proving the assertion or finding a counterexample is very difficult; the supplied status evidence marks it as disproved, although no explicit counterexample is provided here.