Any plumbers in here care to confirm my diagnosis

So I have been battling a backing up floor drain for years now, I snake the drain and not much ever comes out but it always seems to back up when any kind of water is put into the basement sink. I put a bladder in there for 20 minutes, no backups, its only when water goes into the basin sink that the drain overflows.
So in my effort to finally figure this out, I started putting water into the floor drain, and it flows perfectly down, I put the same water into the sink and it backs up along with lots of big bubbles.
This got me thinking, the water is backing up because its not venting well. I went up on the roof and snaked the vent there, still backing up. But, looking at the pipes, I think I found the issue, but I need a plumbers eye (or anyone who knows more about plumbing than me) to confirm my suspicions.

Here are the pipes on the basin sink, you can see the drain pipe from the sink (left), then the next pipe (middle) is the vent, and the far right pipe comes from the kitchen sink. Seems to me that water flowing into the basin sink is causing a vacuum on the vent because as soon as the water stops in the sink, the floor drains fast. I would think that the far right pipe should be the vent, middle for the kitchen sink and left side basin sink. Or at least cap off the middle pipe, and put a T into the right side pipe for the vent.

 

Technically, I believe that is double-vented. The drain it's tying to should be vented. It shouldn't need the middle dedicated vent.

Easy test, plug that vent and see if it's still a problem

 

Floor drain should have its own p-trap and vent. If it’s vent is tied into the sink waste instead of the vent pipe above the sink waste line you’ll have this problem. Another possibility is the wrong connector or pipe size causing sink waste flow to block access to the main drain line for floor drain.