Wh-adverbs and ambiguity

Related to this thread but separate.

(1) Иван КАК пишет диссертацию?
    Ivan KAK pishet dissertatsiju?
    Ivan how writes dissertation
 "How is it that Ivan is writing a dissertation?" [rus]

I have currently the following ambiguity:

(2) Screen Shot 2020-04-27 at 4.49.30 PM

I don’t think this is legit, even though for a non-wh adverb perhaps we would say it is?

(1) Иван долго пишет диссертацию?
    Ivan dolgo pishet dissertatsiju?
    Ivan long writes dissertation
 "Ivan is writing a dissertation for a long time." [rus]

Generally I think we often say that adjunct attachment often comes with annoying ambiguity but in this particular case, I am wondering if this should be handled in the lexical entry for HOW? Perhaps there is something about HOW (and perhaps even generally wh-adverbs) that it simply doesn’t make sense to say that it can modify a verb without its complements?

What would be some tests here?

Also, of course I cannot say that wh-adverbs modify VPs generally, because then I lose:

(1) Иван пишет  диссертацию   как?
    Ivan pishet dissertatsiju как?
    Ivan writes dissertation  how?
 "How is it that Ivan is writing a dissertation?" [rus]

But perhaps I could have a type that is like clist (which cannot contain gaps) except it can contain gaps or be empty but cannot contain canonical synsems? Is that possible?

I think this falls into the category of ‘live-with-it’ spurious ambiguity (multiple trees, same MRS).