The conjecture for extensions beginning with n1 and n2

Let $L(w)$ denote the lexicographically least infinite word on $\mathbb{N}$ beginning with $w$ whose only square factors are contained in the prefix $w$. The conjecture for extensions beginning with $n1$ and $n2$. For every integer $n\geq 3$,

$$L(n1)=nL(1)\qquad\text{and}\qquad L(n2)=nL(2).$$

These identities would extend the known formulas for prefixes $nn$ and $n_1n_2$ with letters at least $3$, and would describe the least square-free extensions of the mixed prefixes $n1$ and $n2$.