Here is a real case I have met in my workplace.
Staff A: Is the goddamn pigeon dead?
Manager: What's happened? (turn to me)
Me: ...Staff A asked me to check if the annoying pigeon is dead.
I am not sure whether it is grammatically correct to use present tense in the above reported speech, becasue some English teacher said this:
But, if the reporting verb is in the past tense, then usually we change the tenses in the reported speech.
Should I say this instead?
Staff A asked me to check if the annoying pigeon had been dead.