Well, that's a bit of a conundrum!! I wouldn't do anything until fall, as soon as they lose their leaves. And I'd make a choice on which one you would prefer to keep, as the process of root disentanglement is not likely to be easy is and probably going to stress both of them. I envision some serious hacking needing to take place so I'd pick one over the other to be more careful with.

I know what my choice would be. I have never been a fan of ROS in any form. But this is your choice ;-)

I'd cut the ROS off at the base, apply full strength (non-diluted) glyphosate very carefully to the stub, and call it done (use a dropper, empty mustard container, etc. to drip it on). Even small ROS seedlings don't pull out easily, and I think by the time you would dig and untangle the roots, you risk both plants.

I use triclopyr concentrate on cut stems- nothing survives that.

RoS are a dime a dozen around here and are considered weeds- I'd treat it like one.

Rose of sharon is an invasive plant. There are sterile forms but yours looks like a seedling deposited by a bird and it's going to be seedy and weedy. Please kill it for your own sake, for your neighbors and for any nearby wooded areas.

ROS is invasive in some places.. mostly because it spreads its seeds too mucgh ... like in the middle of a valuable W ... and odds are.. some bird stopped at the W after eating a seed and planted it there.. im not buying your theory about the prior owner ...

the best day in my garden life.. and the first house was the day i finally killed the prior owners last problem plant.. why it took me 5 years is beyond me...

do what ruth said.. and save yourself 8 hours of backbreaking labor.. just to kill both plants ...

ken

cearbhaill - do you think the triclopyr works on cut tree stems better than glyphosate concentrate? I have both, and since I spend a good chunk of time each season dealing with saplings, I want to be sure I'm doing what works best.

Triclopyr is better on woody material - glyphosate on herbaceous plants.

Interesting to read your comments and I am quite surprised by your apparent general dislike for R.O.S., although I do understand the sucker issue with them and that the dropping blooms can create a mess. Regardless, I have always liked R.O.S, as they provide continuous blooms from mid summer till frost time. My mother had three bushes in three different colors, each was surrounded by a grassy area, and she never had any problems with seedlings. I had two on my previous property where I surrounded the shrubs with heavy groundcover. Seedlings still did pop up, but I always had plenty of takers for them.

In a hedgerow, plant tall weigela varieties with other large shrubs, such as lilacs and viburnums, or on the corner of a building so to not block a view. In a mixed shrub border, plant compact varieties with interesting foliage in with perennial flowers or along the foundation.

The above non sequitur is a cut and paste from https://www.gardeningwithcharlie.com/grow-care-for-weiglea/. (Note misspelled name.)

I’m not sure of their purpose but this poster specializes in this.

Anybody that wants one can come to my house and get ten for free- they pop up in my wooded area like dandelions in a lawn. I don't even know where a mature, flowering plant even is left- I've removed all on my property but they're still coming in via birds.

^^^ Yes, I yank dozens of them annually thanks to the birds. ROS is a terrible plant in the mid Atlantic. I'm glad sterile varieties have been introduced.