English is one of the worse languages to consider mood in, so this is a total misunderstanding.
In a more inflected language like Latin or older forms of Greek the 'is' or the 'has' would have a subjunctive or optative inflection that indicate which of them is more highly hypothetical and therefore to which the necessity applies.
Although Germanic languages sort of try to have a subjunctive, they fail. Instead, mood is not expressed in a verb form, but with auxiliaries or adverbs, and it is assumed to be an attribute of the most primary verb from which it is not fully isolated. English has very weak isolation, so the mood almost always applies to the primary verb and selects a range of possible worlds for the whole sentence to be true or false in.
"I might be going to the store later" can mean that I am going somewhere later and that place might or might not be the store; or that I might or might not be going anywhere at all later, but if I am going, where I am going is to the store. It remains ambiguous because the modality is expressed with its own verb, just tossed in there independently, without being bound to anything.
It is unlikely to mean that I might or might not be going and where I am going, if I go, might or might not be the store, because there is still only one modal verb. To be that ambiguous, we have to say more. How silly is that? Luckily, the exact intention seldom matters to the hearer, who just gets the vague sense of uncertainty, if they care at all.
Beyond that, unlike languages with more hierarchical grammars, English just automatically moves attributes wherever they make the most sense. If I say "tomorrow he says the car will run", we totally forgive the illogic and put the 'tomorrow' on 'run', whether it is grammatically possible or not, because "tomorow he says..." is illogical and "tomorrow, he says, ..." is not.
So there is absolutely no natural way to say the equivalent of "p implies that necessarily q" and get heard that way consistently, because even when it is written just that way, the 'necessarily' automatically jumps back to the 'implies' and the sentence means "p necessarily implies that q", which is "box (p arrow q)". However you try to keep it bound, our speaking habits let it loose and it goes where it is most likely to belong.
You have to descend to indirect reference: "p arrow box q" means "p being true (in this reality) implies q is true in all possible (related) realities".
To some degree that seems just silly to even think about for many English speakers. "How can this reality affect all related realities, sentences hold in realities or they don't..." because moods generally are not that finely articulated in our language.