Second order intuitionistic propositional logic of the real line is decidable

It is known that the set of tautologies of second order intuitionistic propositional logic, $\mathrm{I}\mathrm{P}\mathrm{C}2$, is undecidable. Here, we prove that the sets of formulas of $\mathrm{I}\mathrm{P}\mathrm{C}2$ which are true in the algebra of open subsets of reals or rationals are decidable.