Dilated Floor Functions That Commute

We determine all pairs of real numbers $(\alpha, \beta)$ such that the dilated floor functions $\lfloor \alpha x\rfloor$ and $\lfloor \beta x\rfloor$ commute under composition, i.e., such that $\lfloor \alpha \lfloor \beta x\rfloor\rfloor = \lfloor \beta \lfloor \alpha x\rfloor\rfloor$ holds for all real $x$.