The conjecture that continued-fraction sets with matched bounds cover modulo 1

For $k\in\mathbb{N}$, let $F_{\leq k}$ and $F_{\geq k}$ denote the continued-fraction sets whose partial quotients are respectively bounded above by $k$ and bounded below by $k$. 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 continued-fraction upper-and-lower conjecture. For every $k\in\mathbb{N}$,

$$F_{\leq 2k}+F_{\geq k}\equiv\mathbb{R}\mod 1.$$

The directly matched sum $F_{\leq k}+F_{\geq k}$ has diameter less than $1$, so it cannot satisfy the corresponding covering statement. This alternative conjecture is presented as an analogue of the Lüroth-set results and relates to decompositions involving badly and well approximable numbers.