Plumbing Setup for Basement Laundry

You need have to have a vent between the washer and kitchen drain and you can't have a running trap in the pipe like that either, I know codes are different all over the U.S. but this is the way we would have to do it according to southeastern builders/plumbing code, been a master plumber for 18 years. Not sure if my drawing is going to post but you need a combo for the ks drain and a combo for washer drain, with a trap and pro vent above the trap on a make shift vent terminated wherever you cab access it in the future. Good luck

If it were me, I'd install a new sanitary tee in the main drain, just above the existing sanitary tee. I'd have the standpipe for the washer come down into a P-trap, and connect the other end to the new tee.

However, you'll have to make sure the standpipe is at least 18" above the trap weir. If that height cannot be maintained, you'll have to move some things around.

I'm also not sure if a proper P-trap will fit in the stud bay, so you'll have to measure to verify. The minimum length of a trap arm is two pipe diameters. So if you're using 2" pipe, that's 4" from trap weir to the sanitary tee.

Then I'd have the trap from the washtub, connect to the existing sanitary tee.

Basically, something like this...