Which modal is correct when attached to a present perfect construction?

Question

In the following sentence are may and might interchangeable? There is no question of permission here, only possibility. The modal leads into a present perfect construction, which means the action begins in the past. It's been noted in other discussions that might is the past tense of may, but that is usually met by the denial that modals have a past tense. Are both may and might indisputably grammatically correct in the sentence?

The police then grill Philip about his involvement with the young woman and imply that the fight may (might) have been a setup.

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Answer 1

It is all about dialects, two of them to be precise: A and B.

Dialects A and B are differentiated by reference to constructions like

1. I thought it might rain before we got home.
2. I thought it may rain before we got home.

In the older Dialect A (which I speak) [2] is ungrammatical (just like *"I thought I can finish the book before I got home"): [1], with "might" is required.

In Dialect B, [2] is possible as well as [1]. In Dialect A, "might" is undoubtedly the preterite counterpart of "may", just as "could" is of "can" because it is the form required in backshift. In Dialect B there's no basis for retaining (from earlier stages of the language) the analysis of "might" as the preterite of "may": it must be a distinct lexeme.

I realise that you’re not asking about permission here, but one factor facilitating this linguistic change is that “might” even in Dialect A is hardly used in the primary sense of the preterite, to indicate past time: we usually say "was/were allowed" rather than "might" for past time permission, e.g. "He told me I/we might go").