Realizability conjecture for unitary 2-shaded planar algebras by multifactor inclusions

Let $\cP_\bullet$ be a unitary 2-shaded planar algebra. A finite-index homogeneous connected $\rm II_1$ multifactor inclusion is an inclusion $A\subset B$ with finite Jones index and the stated homogeneity and connectedness properties; its standard invariant is the associated planar algebra invariant. Realizability conjecture. There is a finite-index homogeneous connected $\rm II_1$ multifactor inclusion $A\subset B$ whose standard invariant is $*$-isomorphic to $\cP_\bullet$. The conjecture extends the known realization result for connected spherical unitary 2-shaded planar algebras, where the corresponding inclusion can be chosen extremal, and in finite depth even hyperfinite. It asserts surjectivity of the planar-algebra realization map in general.