The 6n complexity conjecture for binary generalized pseudostandard words

Let $\mathbf u$ be a binary generalized pseudostandard word, and let $\mathcal C_{\mathbf u}(n)$ denote its complexity, namely the number of factors of length $n$ in $\mathbf u$. The 6n complexity conjecture.

$$\mathcal C_{\mathbf u}(n)<6n$$

for every $n\in\mathbb N$. The conjecture is motivated by computer experiments and proposes a new upper bound after the previously stated $4n$ bound was disproved. Determining such upper bounds is presented as an open problem, and no resolution of this conjecture is given in the source.