Italics Formatting Question

I'm working on a sci-fi piece that has a ship called the Priam. Following tradition, when it's mentioned by name I italicize it. My question is that if the name is italicized at the end of sentence, does the following punctuation also become italicized? E.g. She asked, "What is it about the Priam?" or, She asked, "What was it about the Priam?"

That's what I thought too, but it looked... off. Thank you for confirming for me.