special relativity - Does a ticking watch have more mass?

In the video on YouTube, The Real Meaning of E=mc² | PBS| Space Time Studios, it claims that a ticking watch has more mass then a non ticking watch due to the intrinsic KE, PE and thermal energy of the watch's internal movements that manifests itself as part of the watch's mass.

Is this correct ?

To me there is something wrong with this argument, but I cannot put my finger on it.