Is the Tashahhud (testimony of faith) obligatory after the Prostration of Forgetfulness (Sujud Al-Sahw), and what is the authenticity of what was reported in Sahih Al-Bukhari regarding not performing the Tashahhud thereafter?

The ruling on doubting a deficiency in a واجب (obligatory act) varies according to the terminology of the jurists. If the واجب is one of the pillars (أركان) or obligatory acts (فرائض), then the prostration of forgetfulness (سجود السهو) does not compensate for it, and it must be performed. However, if the واجب is not a fard (e.g., confirmed sunnahs of prayer), then it is compensated for by the prostration of forgetfulness.