They can ban every account in the whole world if they want to. They don't need any proof of anything, ban reason can be "fhfhsh".
Although, I have a feeling it was just a poor choice of words, so, I'll go with what you really meant which was "If I get banned on one, do you think they'll both looking into it enough to find my second account?". Answer to that would depend, if both accounts were made on a private ISP IP and were both botting on a private ISP ip and they were the only two accounts on the private ISP ip and they've been on it for a good 6 months, they'd probably look into it. Other than that, they probably won't because of shared IPs/dynamic IPs/etc.
Even if all the conditions above were met, still, unlikely that they'd look into it.