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The phrase 'should he have' is correct and usable in written English
It is used to express something that might have happened, but didn't. Example: I wonder if he would have been successful if he had tried harder - should he have tried harder?
Exact(60)
Should he have known?
Should he have more?
Should he have waited?
Nor should he have many.
Why should he have been?
Should he have said it?
Should he have?
Should he have done?
But should he have?
But should he have been?
Should he have, say, six Theoretical Nominations?
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com