Too many people focus if they have the "relevant skills" while the real important thing for new grads is "excellence in any quantitative field regardless if it has any applications to what the firm does. We don't care too much about relevent experience for a new grad (which you'll be considered as) because to be honest most research is actually not too applicable, what's more important is demonstrating a thorough research process. What HR mainly cares about is brand name (which you sort of have with your T4 PhD degree), and if you interned/worked at a top firm (doesn't have to be a quant finance firm particularly, good tech firms like OpenAI work too). What we mostly care about is if your publications are interesting and if you published in top journals. The extent that coding ability is valued actually depends quite a bit on the firm.
That being said, I think you will get some interviews to these firms. Just be cautious that PhD applicants from the schools you listed are quite common so you need to really know your shit as you will obviously be asked about your research, as well as be able to answer the typical QR-type interview questions (you'd be surprised how many PhDs kind of just suck at more math contest type probability questions that are more about cool tricks and intuition rather than fancy math or rigor, we don't care if you know your Lie algebra, Rough Path or whatever, at some shops we don't even care if you know the ins and outs of
C++, they'll just get the developers to productionalize it while the QRs prototype and datamine).
Some other important things to keep in mind. The stuff you do in QR is not as sexy as a lot of people think it is. We have a problem, you solve it, we don't care about the most interesting or theoretically beautiful solution, we care about efficient and scalable solutions that make us $. End goal of a quant finance firm is obviously make the firm more $, not to publish some paper. You will also not have nearly as much freedom as you have in academia in what you want to work on (flip side, we'll pay our QRs 550k+ USD TC fresh out of school, which I don't think even tenured professors reach with 10x the YOE, that's why some shops like Radix can poach many math professors). Also do some more research about the firms you're applying to. Jane Street and Jump is a whole different business than Two Sigma, the research in prop shops does diverge a bit from research needed in hedge funds. Even though JS and Jump are both prop shops, what they do and what they're specialities are in is quite different.
