> I don't think working out increases testosterone

You think wrongly, they're definitely linked:

"Exercise: Resistance training increases testosterone levels acutely, however, in older men, that increase can be avoided by protein ingestion. Endurance training in men may lead to lower testosterone levels." [0]

[0] https://en.wikipedia.org/wiki/Testosterone#Regulation

>however, in older men, that increase can be avoided by protein ingestion.

Why on earth would you want to avoid it? That's like saying weight loss could be avoided by eating lots of cake.

It's an interesting question, I'm not sure. This is mentioned under the Health Risks section though:

"In people who have undergone testosterone deprivation therapy, testosterone increases beyond the castrate level have been shown to increase the rate of spread of an existing prostate cancer."

So there's probably myriad scenarios where one needs to balance testosterone levels with maintaining physical activity/strength training.

if we included testosterone supplements I think we'll find a correlation

Taking testosterone increases testosterone? You must be joking

the correlation is with the gym

Correlating gym-going with testosterone supplementation?