Is "has or will read" grammatical?

Solution 1:

The relevant grammatical rules involved here are

1. The Perfect auxiliary have must be followed by the past participle form of the next verb.
2. Modal auxiliary verbs like will must be followed by the infinitive form of the next verb.
3. Conjunction Reduction optionally deletes the first of two identical verbs following auxiliaries.

The question is what counts as "identical" for conjunction reduction. And the answer is that "identical" means "identical in sound". Nobody would ever say this sentence, for precisely the reasons described in the answers and comments here. That is, this isn't a question about English; this is about English spelling and reading, which is technology, not linguistics.

The problem with this sentence is that it looks like it's OK, but it doesn't sound like it.
Take a verb like sing, sang, sung, with different infinitive (sing) and past participle (sung) forms.
Then both

• *He has or will sing that song
• *He has or will sung that song

are ungrammatical, no matter which form is used.

And that's why

• *He has or will read that book

is ungrammatical. It could only happen in writing; it's a cheat, like a sight rhyme. It really should be

• He has red or will reed that book
  (spelled funnetikly)

because words pronounced differently can't do conjunction reduction.
And spelling doesn't count.

• No English grammar rule has anything to do with spelling or punctuation.

If you try verbs with identical infinitive and past participle forms, like the set of
monosyllabic final-t verbs like set, set, set; cut, cut, cut; or put, put, put:

• He has or will set the plan in motion.
• He has or will cut them some slack.
• He has or will put it on display in the main gallery.

These sound perfectly grammatical (if needlessly complex), to me. This despite the facts that

• the set, cut, or put following will must be an infinitive,

but

• the set, cut, or put following has must be a past participle.

The abstract grammatical category of the deleted verb seems to be irrelevant -- as long as they sound the same, they're identical. And as long as that's the case, you can delete the first one.

Solution 2:

I must say I couldn't even understand what "has or will read" even meant for a minute or so. I had to skip the phrase itself and read further into the context to begin to understand what you were talking about.

My answer, therefore, is that I am not merely uncomfortable with the deletion, I find it incomprehensible. "Has or will read" for me is simply unacceptable.

For what it's worth, here is something which might shed some light (or at least some interesting color) on the matter. Neurophysiologists have found that if you have two friends named Gandalf, the wave pattern generated in your brain is consistently the same for each one, and not at all the same as the other, even though the words sound the same to the ear and look the same to the eye. In other words, "read" and "read" are very much different words, their visual appearance notwithstanding.