The conjecture that matched upper- and lower-bounded Lüroth sets cover modulo 1

For $k\in\mathbb{N}\setminus\{1\}$, let $L_{\leq k}$ be the Lüroth set with digits at most $k$, and let $L_{\geq k}$ be the Lüroth set with digits at least $k$. Write $A+B=\{a+b:a\in A,b\in B\}$ and $A\equiv\mathbb{R}\mod 1$ when $A+\mathbb{Z}=\mathbb{R}$. The conjecture. For every $k\in\mathbb{N}\setminus\{1\}$,

$$L_{\leq k}+L_{\geq k}\equiv\mathbb{R}\mod 1.$$

This would strengthen Theorem 4, which proves the same assertion with $L_{\leq k+2}$ in place of $L_{\leq k}$. The conjecture is motivated by the observation that the diameter of $L_{\leq k}+L_{\geq k}$ equals $1$, but the source leaves its truth open.