Non-isomorphism of ultrapowers for structures with the independence property

Let $M$ be a structure whose first-order theory has the independence property witnessed by an affine quantifier-free formula. Let $(\Omega,\mu)$ be a diffuse probability space, and let $\mathcal{U}$ be a countably incomplete ultrafilter. For a diffuse measure space $\Lambda$ and a structure $N$, consider the ultrapower $L^1(\Omega,M)^{\mathcal{U}}$. Non-isomorphism conjecture. $L^1(\Omega,M)^{\mathcal{U}}$ is not isomorphic to $L^1(\Lambda,N)$ for any diffuse measure space $\Lambda$ and any structure $N$. This is suggested by the proof of the preceding proposition, which gives evidence that the independence property obstructs such an isomorphism; the source does not provide a resolution.