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user65526
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Can we avoid the modal collapse in a certain Intuitionistic modal logic by abandoning ¬◯⊥ but retaining the law of the excluded middle?

A proof of the modal collapse (We can derive both $\bigcirc M \supset M$ and $M \supset \bigcirc M$) in a certain Intuitionistic modal logic was given here: Modal collapse upon addition of the law of the excluded middle to an Intuitionistic modal logic.

The modal collapse was claimed in https://www.uni-bamberg.de/fileadmin/uni/fakultaeten/wiai_professuren/grundlagen_informatik/papersMM/pll.pdf:

"...if we add the axiom of the Excluded Middle (EM) and $\neg \bigcirc false$ which is valid for both $\Diamond$ and $\Box$ to the modal system $\bigcirc R, \bigcirc M, \bigcirc S$ then $\bigcirc$ becomes trivial. We can derive both $\bigcirc M \supset M$ and $M \supset \bigcirc M$ In other words there is no classical Kripke semantics for $\bigcirc$." (p.4, para 1 of the above article)

However, I was wondering

(1) Can we get the modal collapse if we only assume the law of the excluded middle, and we don't assume $¬◯⊥$?

(2) If we are able to avoid the modal collapse by abandoning $¬◯⊥$, would abandoning $¬◯⊥$ be problematic in other ways?

user65526
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