Is it obligatory to pray تحية المسجد (two rak'ahs upon entering the mosque) after the adhan has finished, or is it permissible to pray it during the adhan?

It is recommended for someone who enters the mosque while the muezzin is calling the adhan to respond to the muezzin, then supplicate with the reported supplications, and then pray Tahiyyat al-Masjid (two rak'ahs upon entering the mosque). Some scholars made an exception for those who enter the mosque on Friday while the muezzin is calling the second adhan, stating that they should pray Tahiyyat al-Masjid in order to listen to the sermon. This is because listening to the sermon is obligatory, while responding to the muezzin is not, and the obligatory takes precedence. Others are of the view that one can combine responding to the adhan with praying two light rak'ahs.