The propositional theory conjecture for the Muchnik lattices at levels 2 and 3

For $n\in\{2,3\}$, let $\mathcal{M}_n$ denote the corresponding Muchnik lattice, and let $\mathrm{Jan}$ and $\mathrm{IPC}$ denote the propositional theories of Jan and intuitionistic propositional logic, respectively. A principal factor is a quotient of $\mathcal{M}_n$ by a principal congruence.

Propositional theory conjecture. The propositional theories of $\mathcal{M}_2$ and $\mathcal{M}_3$ are $\mathrm{Jan}$; in fact, there are principal factors of $\mathcal{M}_2$ and $\mathcal{M}_3$ which have propositional theory $\mathrm{IPC}$.

The cases $n\leq 1$ and $n\geq 4$ are handled in the paper, with $\mathrm{Th}(\mathcal{M}_n)=\mathrm{Jan}$ established for $n\geq 4$. The cases $n=2$ and $n=3$ remain unresolved in the supplied text; the stronger assertion about principal factors would realize intuitionistic propositional logic in those levels.