The explicit limit 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 $w$. Define the finite words $c_n$ recursively as in the paper, and let $\gamma$ be the morphism defined from the words $R_n$ and $b_n$ by

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

and, for $n\geq 0$,

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

The explicit limit conjecture for $L(2)$.

$$L(2)=2\lim_{n\to\infty}c_n=2\gamma(L(\varepsilon)).$$

This is the explicit version of the proposed structure of $L(2)$; it identifies the conjectured suffix both as the limit of the recursively constructed words $c_n$ and as the image of $L(\varepsilon)$ under $\gamma$.