The name was given as "SHI-KE XUE", and the calligrapher is asking if it should be "Shi-Ke Xue" or "Shi-ke Xue". As far as I know, there is no context about where this person is from, what chinese language they speak, or anything other than the name. I think the calligraphy is for a swanky invitation, so it might not be possible to talk to them beforehand.
I thought it would be relatively straightforward (figure out if the original romanizer used pinyin or Wade-Giles and apply the rules!) but it seems more complicated than that, because (AFAIK, not knowing Chinese) there aren't hyphens in pinyin and there aren't "x"s in WG. In either of the main two systems, though, the second syllable of a compound name would be lowercase, right?
I said that I was totally unqualified to answer, but my best guess was "Shi-ke Xue".
So, sinolinguaphiles: Is there a way to know how to write this in Latin letters that doesn't involve getting more information? Is there some sort of best practice to use when writing a romanized Chinese name if you don't know the person or how they usually spell their name? If you have a Chinese name and someone capitalized it differently than you normally would, would you be offended?