Laundry Sink into Washer Drain Stand Pipe

This has been occupying a lot of my time... and I thank you all for your help so far!

I am thinking of going back to my original plan to put a laundry sink closer to my washing machine. Not knowing much about plumbing, originally I was simply going to "T" into my washing machine stand pipe. Thankfully, someone warned me about siphoning, sewer gas, etc.. Though the attached "plan" might not be ideal, would it work? Am I right in thinking that I wouldn't have to care if this setup siphoned my sink's p-trap since there is a trap below that will block sewer gas? To me, it doesn't seem any different than the current setup of having the washing machine drain hanging into the top of the open stand pipe. What am I missing in my way of thinking?

Thanks again for all of the advice offered!

Two traps never works well...no that would not work satisfactorily. And, you might end up with water in the sink when the WM drains.

Each fixture must have it's own trap. Each trap must have a vent.
In WI it is required that the washer trap be above the floor.
Our code is not too much different than in other states.

Just do away with the trap under the sink since you are going into the stand pipe anyway and it will work and you can live with it no matter how it looks or smells