Should the worshiper prostrate for the prostration of recitation if he recites an verse containing a prostration secretly in a silent prayer, such as Asr, while praying with the congregation, or should he not prostrate and continue his prayer with the congregation?

It is not permissible for the congregant to perform the prostration of recitation (sujud at-tilawah) except by following their Imam; otherwise, their prayer becomes invalid. This is based on the Prophet's (peace be upon him) saying: "Indeed, the Imam is appointed to be followed, so do not differ from him."