I am very grateful to Vijay Fafat for bringing this excellent work of mathematical fiction to my attention and writing a nice review to post here. My viewpoint differs in just a few places, so please allow me to comment.

Vijay is right that it would be ridiculous to completely rule out discussing one conjecture while attempting to prove another. Doing so is not necessarily "desperate" or "grasping at straws". In particular, it should be noted that Andrew Wiles proved Fermat's Last Theorem by proving the Taniyama-Shimura-Weil Conjecture. So, clearly the consideration of other conjectures can sometimes be useful when attempting to resolve a famous open problem. On the other hand, I did not find the (digitally simulated) father's objection to be unbelievable either. When there is an open conjecture, mathematicians in the field will each have their own opinions both on how likely it is to be true and how hard it will be to prove. If the elder Gifford in the story thought either that the Jagdish-Rajput Conjecture was not likely to be true or thought that it would be harder to prove than the Perelman Hypothesis itself, he might well have responded exactly in this way.

One of my favorite things in mathematical fiction are metaphors in which a mathematical idea is used to convey something deep about the human experience. As Vijay has hinted, the notion of proof by induction shows up in the story in several places, including metaphorically in the multi-generational family dynamics. For me, even though this is a short story and there wasn't much room to elaborate on it, the idea was still potent and effective. In fact, it may even have worked better because the author didn't have room to do much with it, leaving it up to the reader to fill in the gaps.

That the names Perelman and Ricci are mentioned along with the word "hyperbolic" certainly suggests that the author had differential geometry in mind as the subject on which the two Giffords were collaborating. However, there is nothing explicit in the story which says so. Similarly, there is nothing that actually says that the "Perelman" whose hypothesis they are working one, is Grigori Perelman. Perhaps I'm being like the mathematician from the joke ("No, we only know that there is one sheep in this country which is black on one side!") but it seems like it could be someone else with the same name.

This story was published in the May/June 2021 issue of Uncanny Magazine and is available for free online at their website.