Kim's pseudolinearity conjecture

Let $T$ be a simple theory, and let $p\in S(A)$ be a Lascar type. The type $p$ has Lascar rank $\mathrm{SU}(p)=1$, is $k$-linear when it satisfies the corresponding $k$-linearity condition, and is linear when it satisfies the corresponding linearity condition.

Kim's pseudolinearity conjecture. If $\mathrm{SU}(p)=1$ and $p$ is $k$-linear for some $k<\omega$, then $p$ is linear.

This conjecture concerns the relationship between finite linearity conditions and linearity for rank-one types in simple theories. The supplied text gives no resolution status.