Divisor function monotonicity conjecture

For $r>1$, let $C_r$ denote the number of connected components of the closure of the image of the regular divisor function $[?]$. Divisor function monotonicity conjecture. $C_r$ is monotone in $r$. The source presents this as an analogue of a conjecture for unitary divisor functions; no resolution is given for the regular divisor-function version.