"Aliyun waliullah" in tashahhud, obligatory?

I had earlier made a post (click here) discussing whether reciting the third tetimony (aliun waliullah) or anything other dhikr after the shahadatain (the two testimonies) makes the prayer invalid, but does that mean that saying the third testimony (aliun waliullah) in prayers is obligatory in any way? What do our authentic ahadith say?

Muntaqal Jiman

باب التشهد والتسليم

Section on tashahhud and taslim

صحي: محمد بن الحسن، بإسناده عن سعد بن عبد الله، عن العباس بن
معروف، عن علي بن مهزيار، عن حماد بن عيسى، عن حريز بن عبد الله، عن
زرارة قال: قلت لأبي جعفر عليه السلام: ما يجزي من القول في التشهد في الركعتين
الأولتين؟ قال: [أن] تقول: (أشهد أن لا إله إلا الله وحده لا شريك له)
قلت: فما يجزي من تشهد الركعتين الأخيرتين؟ فقال: الشهادتان

Sahih (Correct/Authentic): Muhammad b. al Hasan, by his isnad (chains of narration) from Sa'd b. Abdullah, from al Abbas b. Ma'roof, from Ali b. Mahzhar, from Hammad b. 'Isa, from Huraiz b. Abdullah, from Zurarah who said: "I said to abi Ja'far (as): What suffices of the saying in the tashhud in the first two rak'ahs?" He (as) said: "[That] you say: Ashhadu anna la ilaha illa Allah, wahdahu la sharika lahu (I testify that there is no deity except Allah, He is one with no partner for him.)". I said: "So what suffices of tashahhud in the last two rak'ahs?" So he (as) said: "The two testimonies."

وبإسناده عن أحمد بن محمد، عن أبيه، عن أحمد بن محمد بن أبي نصر، قال: قلت لأبي الحسن عليه السلام: جعلت فداك التشهد الذي في الثانية يجزي أن أقوله في الرابعة، قال: نعم

And by his isnad (chains of narration) from Ahmad b. Muhammad, from his father, from Ahmad b. Muhammad b. abi Nusr, who said: "I said to abi al Hasan (as): May I be your ransom, the tashahhud which is in the second, it suffices that I said it in the fourth?" He (as) said: "Yes."