But it seems that the following statement is a counterexample.
Ψ: It is impossible to know Ψ.
Claim. Ψ is true (and, thus, it is impossible to know Ψ).
Proof. Assume (to reach a contradiction) that Ψ is not true. Then it is possible to know Ψ. So, there is a world w such that Ψ is known at w. Since Ψ is known at w, Ψ is true at w. But then it is impossible to know Ψ at w. Hence, (assuming T or stronger) Ψ is not known at w. Contradiction.
[Does this work? It is just a "modalized" version of the Knower. Has this already been done?]