Independence conjecture for cycle indicators of $b_r$-regular permutations

Let $r>2$ and let $b_r=[1,\ldots,1,2,\ldots,n-r+1]$ be the restriction vector with $r$ initial entries equal to $1$ and the remaining entries increasing consecutively. A permutation satisfying the corresponding one-sided restriction is called $b_r$-regular. Two cycle indicators are $r-1$-separated when the distance between the numbers their cycles include is at least $r-1$.

Independence conjecture. For $b_r$-regular permutations, any two cycle indicators are independent if and only if they are $r-1$-separated.

This conjecture extends the corresponding $r=2$ independence property to the wider family of one-sided restrictions. Establishing it would help control the more complicated local dependence structure needed for central limit theorems for cycle counts when $r>2$.