Does the prayer of one who intended to make up the Dhuhr prayer along with the Asr prayer become invalid if he forgot during the prayer that it was Dhuhr and thought it was Asr, then remembered before the Tasleem that it was Dhuhr?

The jurisprudential schools mention that there is a disagreement regarding the invalidity of a prayer performed with an incorrect intention. Imam Ahmad was asked about an Imam who prayed Asr, thinking it was Dhuhr, and prolonged it. He replied, "He should repeat [the prayer]." However, the author of Al-Furu' mentioned a view that one who concludes a four-rak'ah prayer, thinking it is the Friday prayer, does not need to repeat it. In our case, it is preferable to repeat the prayer as a precaution, but if he does not, we hope it will suffice him.