Very much the latter
Without digging too in-depth, NJ's World League (G-1 precursor in the 70s/80s) seemed to bounce around the year so it may have had overlap with the CC at some point, but since 1991 the G-1 has always been July/Aug and CC has always been in March/April
The first Global League in 2010 was also in March/April and did compete with the CC that year, but after that they moved it to Oct/Nov until the N-1 rebranding (which I still hate) last year when NOAH pulled it back to September, so that's the one tourney that would have been happening right now under normal circumstances
Fake edit: Oh yeah, did forget pre-COVID that the G-1 was going to take place at this time anyways because of the Tokyo Olympics