CC interfering with CS6 Registration

I have a subscription for the full Creative Cloud.  I use some of the CC apps on my Mac laptop.

I also have been a longstanding user of Creative Suite 6 (CS6), and I still use those applications on my home desktop computer.

Both the CC and the CS6 products are registered to my same email address.

The past few times when I have tried to use my CS6 products on my desktop, I get error messages telling me that they are either in a Trial Period, or that I am not authorized to use them.  (Illustrator, InDesign and Photoshop told me I was in a trial period, and Acrobat Pro told me that I was not authorized.)

I restarted each of those products, and they appear to be working ok now, but I am concerned about this behavior.

For clarity, the I have not installed any CC apps on my desktop where i had been using CS6 for years.  It appears as though my account un-registered the CS6 products without me doing anything.

Is there an issue?  If I ever stop subscribing to CC, will I have further issues with CS6 products that I bought?

I tried contacting Adobe Support about this, and the first rep seemed to confuse CS6 with older unsupported versions of CC, and tried to upsell me!  The next support rep told me to contact them again if I experienced.

I'm wondering if anyone here on the forum has had experience with this issue in the past?

Thanks,

Nick