The conjecture that two lower-bounded Lüroth sets contain an interval

For $k\in\mathbb{N}\setminus\{1\}$, let $L_{\geq k}$ be the Lüroth set whose digits are all at least $k$. For subsets of $\mathbb{R}$, write $A+B=\{a+b:a\in A,b\in B\}$. The interval conjecture for lower-bounded Lüroth sets. For every $k\in\mathbb{N}\setminus\{1\}$, the sumset $L_{\geq k}+L_{\geq k}$ contains an interval of positive length in $\mathbb{R}$.

This is proposed as an analogue of Shulga's result after the paper proves that the same sumset has Hausdorff dimension $1$. The source presents the interval-containment assertion as a conjecture, and no resolution is supplied.