The stationary $K$-model conjecture for quantum subgroups of $S_N^+$

Let $G\subset S_N^+$ be a closed quantum subgroup whose orbits all have size $K$, and assume the supplementary transitivity-type condition stated in the source. A stationary model is a matrix model for $C(G)$ realizing the Haar state, and a stationary $K\times K$ model is one whose matrix size is $K$. Let the universal quasi-flat model mean the universal model in which the relevant projections satisfy the quasi-flatness condition.

Stationary $K$-model conjecture. Under these assumptions, the following are equivalent:

1. $C(G)$ is of type I.
2. $C(G)$ has a stationary model.
3. $C(G)$ has a stationary $K\times K$ model.
4. The universal quasi-flat model for $C(G)$ is stationary.

The implications $(4)\implies(3)\implies(2)\implies(1)$ are described as trivial, while $(1)\implies(2)$ is expected without the extra assumptions. The implication $(3)\implies(4)$ is proved under a mild assumption, leaving the substantive implication $(2)\implies(3)$ conjectural.