"engross", "captivate", and "grip" have the shared meaning of attracting and occupying someone's attention.
The emphasis of the words are like so:
- engross: occupies the attention ENTIRELY and CONTINUOUSLY for a while.
- captivate: holds the attention because of an INTERESTING and FASCINATING nature.
- grip: holds the attention with compelling FORCE and URGENCY (for a short period).
Therefore "engross" is the best fit for a prolonged fascination with a TV program.
But as important as the difference in meaning is the preposition "in" which follows the underline.
When referring to the sense mentioned, we usually say "(be) engrossed in", "captivated by", and "gripped by/with".
(when "be gripped" is used with the external cause, "by" is used - "I was gripped by the sight" - not "with", or "in")
So, because of the preposition, captivate and grip are ruled out.
* And yes, "grip" also means physically holding something ("He gripped me by the arm").