This is a problem I came up with on my own, and it has me stumped, so I am going to pose it as a kind of puzzle.
Let $F$ be a formal proof system, recursively axiomatizable, with an acceptable Gödel numbering, and conservatively extended with the Gödel-Löb system of modal logic for provability. Making some reasonable consistency assumptions on $F$, at a minimum that $$\lnot\Box\Box\Box(\varphi\land\lnot\varphi)\tag{1}$$ for any sentence $\varphi$ in $F$, is it possible to construct a sentence $\psi$ such that $$\Box\big(\Box\Box(\psi\land\lnot\psi)\leftrightarrow\Box\psi\lor\Box\lnot\psi\big)\tag{2}$$ holds in $F$?
More explanation: someone asked why I require three levels of consistency and no more no less. The reason is that if (1) fails to hold for some sentence $\varphi$ then it is false for all sentences $\varphi$. In that case, $$\Box\big(\Box\Box(\psi\land\lnot\psi)\big)\tag{3}$$ holds for any $\psi$, so any $\psi$ that satisfies $$\Box\big(\Box\psi\lor\Box\lnot\psi\big),\tag{4}$$ for example, any provable sentence, would satisfy (2) since both sides of the equivalence would then be proven to be true. I really want to assume that $F$ is some system such as PA or ZFC where (1) could obviously be assumed to hold, but here (1) seems to be the bare minimum assumption on consistency that makes the present problem non-trivial.