I do not know whether the Messenger of Allah (ﷺ) would recite the Quran at the noon and afternoon prayer or not.
Sunan Abi Dawud Book 2, Hadith 419
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Abdullah ibn Ubaydullah said: I went to Ibn Abbas accompanying some youths of Banu Hashim. We said to one of them: Ask Ibn Abbas: Did the Messenger of Allah (ﷺ) recite (the Quran) in the noon and afternoon prayers? He replied: No. People said to him: Perhaps he might recite the Quran quietly. He said: May your face be scratched (a kind of curse)! This (statement) is worse than the former.
He was only a servant (of Allah) receiving Commands from Him. He preached (the divine) message which he brought with him. He did not command anything to us (Banu Hashim) specially excluding other people except three points: he commanded us to perform ablution perfectly, and not to accept charity (sadaqah) and not to make pairing of donkey with horse.
Sunan Abi Dawud Book 2, Hadith 418 Next Hadith → I heard Ibn’Abbas reciting wa’l-mursalat urfan(surah lxxxvii). She said; sonny you have reminded me of this surah by your recitation. Thie is the last surah which I heard the Messenger of Allah(ﷺ) reciting in the sunset prayer.
Sunan Abi Dawud Book 2, Hadith 420
He was only a servant (of Allah) receiving Commands from Him. He preached (the divine) message which he brought with him. He did not command anything to us (Banu Hashim) specially excluding other people except three points: he commanded us to perform ablution perfectly, and not to accept charity (sadaqah) and not to make pairing of donkey with horse.
Sunan Abi Dawud Book 2, Hadith 418 Next Hadith → I heard Ibn’Abbas reciting wa’l-mursalat urfan(surah lxxxvii). She said; sonny you have reminded me of this surah by your recitation. Thie is the last surah which I heard the Messenger of Allah(ﷺ) reciting in the sunset prayer.
Sunan Abi Dawud Book 2, Hadith 420