If the verb in the independent clause is in the present tense, in which tense should the verb in the indirect quotation be?

If the verb in the independent clause is in the present tense, the tense that the verb in the indirect quotation should be is remain in its original tense.
You don't have to shift tenses because it is present in the independent one.
For example:
He says: "I need to wash my hair."
He says that he needs to wash his hair.
You wouldn't say - he says that he needed to wash his hair.