True or false: if a modal sentence $\phi$ is consistent in K, then it is consistent in S5.
This is equivalent to the contrapositive: if $\phi$ is not consistent in S5, then it's not consistent in K. This seems to be false because if we can prove $\neg \phi$ in S5, it is not necessarily the case that we could prove it in K.
Here is a counterexample. Let $\phi=\top\land\square\bot$. This sentence is consistent in K because it is satisfiable in K. But its negation $\neg(\square \bot\land \top)$, which is logically equivalent to $\square \bot\to\bot$ is provable in S5.
In particular this proves the stronger statement that a modal sentence consistent in K need not be consistent in KT (an in particular need not be consistent in S5).
Is this counterexample and argument correct?