shakai houken system - own business vs employer's (Tokyo)

I am new to this forum, so I thank all of you in advance for whatever advice and expertise you may be able to give me.

My wife and I have just formed a GK - a goudougaisha, for tax purposes. We've done and submitted all the paperwork for it.

So basically we're good to go on this, but one thing I have to do that we didn't know about so clearly before is that I have to switch my shakai houken over to the corporate scheme. Now the thing is, I am a full-time university teacher, as of recent, and I still have to switch over to the private university social insurance scheme. I've held off submitting the paperwork for that because of the ambiguity of being on the corporate and the private university social insurance schemes at the same time.

I don't want to bore any of you with more details than necessary on that, so here's my main question: for any of you on this forum with full-time jobs who have formed your own companies (either GK or KK) on the side, were you obligated to go onto the corporate social insurance scheme for your GKs or KKs? If so, were you able to be on your employers' social insurance schemes at the same time?

Technically, there is a conflict of interest by being on both at the same time, but I have heard about other people around who have side businesses like these with full-time jobs.

I'd like to network (discreetly), find out more, and possibly meet with some people who have done this. Thank you in advance.