Second-order obstruction conjecture for pulling apart four 2-spheres

Let $X$ be a $4$-manifold and let $A:\amalg^4 S^2\to X$ be a quadruple of immersed $2$-spheres. Suppose that $A$ admits an order $2$ non-repeating Whitney tower $\mathcal{W}$, and let $\lambda_2(A):=\lambda_2(\mathcal{W})$ be the order-$2$ intersection invariant, valued in $\Lambda_2(\pi,4)/\mathrm{INT}_2(A)$. Second-order obstruction conjecture. The spheres in $A$ can be pulled apart if and only if

$$\lambda_2(A):=\lambda_2(\mathcal{W})$$

vanishes in $\Lambda_2(\pi,4)/\mathrm{INT}_2(A)$. Necessary and sufficient algebraic conditions for arbitrary quadruples are not currently known; this conjecture specifies the desired criterion once the order-$2$ intersection relations are formulated.