3 problems

Existence conjecture. There exists a non-almost-semi-inner automorphism $\varphi\in\operatorname{Aut}(_{L}\mathcal{M}^{0})$.

Converse conjecture. The converse of Proposition 3.14 is false: not every semi-inner automorphism of $(_{R}\mathcal{M}^{0})$ is equinumerous.

Existence conjecture. There exists such an $R$ having a semi-inner, non-equinumerous automorphism $\varphi\in\operatorname{Aut}(_{R}\mathcal{M}^{0})$.