I think so long as they say it must be independent work and give a time limit then it's fair.

What year of student? Freshmen/Sophmore?

There were a few unsolved conjectures presented on the board on occasions. What was more common, as a first year maths student, was to be presented with homework that had 10 questions on of which one would be marked "hard" and one "very hard". Some of the "very hard" questions would be unsolved research problems.

Some interviewers would present unsolved problems in interviews as well, if a candidate was happily eating their way through the regular material.

Cambridge is a little unusual in that the only mark that people cite is the one on the final year exam. I believe US universities operate by continuous accumulation of marks? And that ongoing disputes over marking are more common because of this?

It's also already working with a highly selected group of people from the far tail of the ability distribution, and looking for a way to identify the exceptional among the merely very high ability.

What you are describing is entirely fair. The student has been fairly warned that they might not be able to solve the problem in the 8-10 minutes that their incredibly packed schedule allocates to it. They aren't being blindsided by an unsolved research problem masquerading as a standard question on a problem set. Fair game.