The morphic structure conjecture for L(2)

Let $L(w)$ denote the lexicographically least infinite word on $\mathbb{N}$ beginning with $w$ whose only square factors are contained in the prefix $w$. Let $L(\varepsilon)$ be the corresponding word with empty prefix, and let $\gamma$ be the morphism defined by

$$\gamma(0)=c_3c_3^+,$$

and, for $n\geq 0$,

$$\gamma(n)=R_4b_4^+\;R_5b_5^+\cdots R_{n+2}b_{n+2}^+\;R_{n+3}b_{n+3},$$

where the words $b_n$, $c_n$, and $R_n$ are defined recursively in the paper. The morphic structure conjecture for $L(2)$.

$$L(2)=2\gamma(L(\varepsilon)).$$

This conjecture proposes an explicit morphic description of the lexicographically least square-free extension beginning with $2$, analogous to the established descriptions for $L(1)$ and $L(n)$ with $n\geq 3$.