How to prevent washing machine discharge from spraying out of drain?

Since moving into a new house, we’ve been battling with an issue with our washing machine. We’ve got it plumbed in beneath a sink (as the previous owners had done) as shown in the image below:

Above this image is the sink. The red box on the right of the image is where the drain from the washing machine enters the drain. Just above that is a black hose which is connected to the sink’s overflow. The red box on the right is the problematic bit…

When we use the sink, this arrangement seems fine. Water drains quickly without any leaks. When we use the washing machine, however, we often find that there is water pooled on the floor. Having watched the washing machine during the "drain" part of its cycle, I found that the water was being forced up through the other arm of the "Y" (red box on the right).

As you can see in the image, I’ve applied some tape around the problematic fixture as temporary fix. Perhaps not surprisingly, this hasn’t performed that well. In case it’s relevant, the tape is covering a sort of tapered, stepped pipe. The sort that you might push a rubber hose onto.

One other thing I’ve tried is to detach the offending part from the "Y" and cap off the hole. From my local hardware store, I’ve purchased a range of "cap ends", but none of them seem to fit. I think I’d measured the wrong part – my measurements say 40mm dia, but both the 40mm or 32mm caps I purchased seem too large.

Finally, worth noting that I’ve detached the trap / "U" for inspection. It’s not blocked or anything.

I’m now backing up a bit and wondering if I’m going about this all wrong. Is there a correct way to solve this problem?