Pull-apart obstruction conjecture for non-repeating Whitney towers

Let $A:\Sigma\to X$ be a collection of immersed $2$-spheres in a $4$-manifold, and suppose that $\lambda_n(A):=\lambda_n(\mathcal{W})$ is defined using an order $n$ non-repeating Whitney tower $\mathcal{W}$, with values in $\Lambda_n(\pi)/\mathrm{INT}_n(A)$. Pull-apart obstruction conjecture. For every $n=2,3,4,\ldots$, $A$ can be pulled apart if and only if

$$\lambda_n(A):=\lambda_n(\mathcal{W})$$

vanishes in $\Lambda_n(\pi)/\mathrm{INT}_n(A)$. This would refine the known necessary obstruction theory into a necessary and sufficient criterion; the required intersection relations are not yet fully formulated, and the claim is posed as a challenge for future work.