A more interesting question is whether there is a Turing test that is easy for ALL humans to pass, while still being hard for LLMs.
In practice, most of the major CAPTCHA vendors already rely on non-privacy-preserving tests for those needing more accessible solutions than a visual puzzle.
Google's audio captcha (only available in a few languages and unusable for those who also have hearing issues) only works for a narrow band of users, not trusted enough to bypass the captcha entirely, but also not untrusted enough. If you fall outside of that band, you get a nice "your device has been classified as a fraud risk, please use the visual captcha" message.
hCaptcha goes even further and straight-up requires you to have an "accessibility cookie", which requires verifying your email address (and apparently your phone number in some cases) to obtain, as well as disabling some anti-tracking settings in your browser.
I've seen one recently where it's basically a series of animated objects and you're asked to click on the slowest one. It's surprisingly easy as a human, but anything that depends on a single screenshot of the page isn't able to solve it.
Obviously, that's only solveable by sighted humans, not ones that are blind or have otherwise low vision.