The conjecture that bounded-above Lüroth sets with digits 3 and 4 cover modulo 1

Let $L_{\leq k}$ denote the Lüroth set whose digits are at most $k$, for $k\in\mathbb{N}\setminus\{1\}$. For subsets of $\mathbb{R}$, write $A+B=\{a+b:a\in A,b\in B\}$, and write $A\equiv\mathbb{R}\mod 1$ when $A+\mathbb{Z}=\mathbb{R}$. The conjecture.

$$L_{\leq 3}+L_{\leq 4}\equiv\mathbb{R}\mod 1.$$

Theorem 1 establishes the analogous covering results for $L_{\leq 3}+L_{\leq 5}$, $L_{\leq 4}+L_{\leq 4}$, and three copies of $L_{\leq 3}$, but does not determine this mixed two-set case. Numerical clustering suggests that $L_{\leq 3}+L_{\leq 4}$ may contain an interval such as $[1,2]$.