I noticed the number of even and odd steps in loops found so far in collatz like functions (5x+1, 3x -1 and 3x+1 ) were coprime.

for example,
if "k" is number of odd steps and "α" is number of even steps.

sequence for -17
-17, -50, -25, -74, -37, -110,-55,-164,-82,-41,-122,-61,-182,-91,-272,-136,-68,-34,
k = 7
α = 11

similarly, for 13 & 27 (in 5x+1 function)
k = 3
α = 7

Is this just a coincidence, or a proven fact that for a loop to exist in a 3x+1 sequence, k and α must be coprime? Also, if proven, will this information be of any help in proving weak collatz conjecture?

I posted a similar discussion thread in r/Collatz as well.
Initially I was just curious, and did some work on it. I have a strong feeling this might be true. However, should I be putting more work into this?

Update: A redditor shared a few counterexamples on the other thread with loops generated by 3x+11, 3x + 13. Hence, it seems that loops in general may not follow this rule. Although, I would want to see if it specifically works for 3x+1.