I expect at least one of these problems is so unimaginably hard that it eludes a solution for 1000 years. Maybe Hodge or P vs. NP.

I hope...

It is extremely rare for mathematical problems in the age of modern research math to be open for more than a thousand years. The most famous example is whether there are odd perfect numbers, and that's... about it? Fwiw I am not a qualified historian of math.

Therefore, this question is mostly whether intelligent forms interested in the Millennium problems will still exist by year 3000, and how long they will be able to maintain work comparable to the current age.

I would not go below a base rate of ~85 %.

@Lilemont There's also infinitude of perfect numbers, the congruent number problem (just barely over 1000 years), and maybe whether there are infinitely many amicable numbers. All of these are also parts of number theory.