144 hour transit visa for HK - Beijing - Taipei possible?

Your itinerary is definitely eligible for 144 hour visa free transit in wider Beijing area and you do not need any kind of visa. Of course, I am concluding this under the assumption that you do not have any additional landing within mainland China beside Beijing. I confirm that, for the 144 hour visa free transit purpose, Hong Kong and Taiwan are considered as third countries along with Macau.

You will need to fly because the train that currently operate on Beijing to Hong Kong route belongs to high speed trains that use West Kowloon station in Hong Kong and this station is not eligible for 144 hour visa free transit. Only Hong Kong Hung Hom station is eligible, but since the introduction of high speed trains, a conventional through train from Hong Kong Hung Hom station to Beijing was discontinued. Those conventional through trains were eligible for 144 hour visa free transit and they still are on the paper, but it is unrealistic to expect the resumption of their services.

thanks!!!