Is the Hadith: "Do not answer one who initiates speech before greeting" authentic?

The Hadith: "Do not respond to one who speaks before Salam (greeting of peace)" was ruled as invalid by Abu Hatim al-Razi. This Hadith has been narrated through multiple chains from Ibn Umar, Jabir, and Abu Hurairah, all of which are weak or denounced (munkar); none of them are authentic.

Despite this, Salam is considered one of the important etiquettes of the religion and a right of a Muslim upon his brother. Some scholars held the view that initiating Salam is obligatory, based on the Hadith of Abu Hurairah: "The rights of a Muslim upon a Muslim are six: when you meet him, greet him with Salam..." while others held that it is a recommended Sunnah.