The Feit-Thompson conjecture states: If $p<q$ are primes, then $\frac{q^p-1}{q-1}$ does not divide $\frac{p^q-1}{p-1}$.

On page xiii of these proceedings of a conference at the University of Yamanashi (Japan) that took place in October of 2017, a proof of the Feit-Thompson conjecture was announced.

Questions:

I heard that this would greatly simplify the Feit–Thompson theorem on odd order groups. Can someone explain the simplification to someone who only is familiar with group theory on a basic algebra textbook level?

What other implications does the Feit-Thompson conjecture have?