Yes, they do store the value. There was a blogpost posted to HN last year that discusses the cryptographic features of these cards [1,2]. It's amazing that these cards are almost solving the problem of secure digital money.

[1] https://news.ycombinator.com/item?id=43993711

[2] working link since original blog is down https://web.archive.org/web/20250517082035/https://aruarian....