Clarification of leftmost maximal periodicities
Main explicitly states that his algorithm may output some nonleftmost or nonmaximal occurrences of periodicities. Indeed, theorem 3.4 only guarantees that every leftmost maximal periodicity will be reported at least once and that total size of the output is linear in size of input string. Namely,


It may report some nonmaximal periodicity, which is suffix of $t_h u_h$ and still can be extended to the right.
There might be several occurrences of the same substring such that none of them fit into a single s-factor.


Kolpakov and Kucherov slightly modify Main's algorithm to exclude suffixes of $t_h u_h$ but allow either $t_h$ or $u_h$ to be empty. This way the algorithm never reports nonmaximal periodicities and at the same time finds every occurrence of maximal periodicity that either spans several s-factors or touches their boundaries.

So, Kolpakov and Kucherov don't rely on the notion of leftmost occurrences at all. They only need to find every periodicity that is not strictly inside single factor of s-factorization. (And prove that there will be only linear number of occurrences in total, of course.)