Pulling-apart criterion for four components in a simply connected 4-manifold

Let $X$ be a simply connected $4$-manifold, and let $A=A_1,A_2,A_3,A_4$ support an order-$2$ non-repeating Whitney tower $\mathcal W$. Let $\lambda_2(A)$ be the order-$2$ intersection invariant, and let $\Lambda_2/\operatorname{INT}_2(A)$ be the quotient by the intersection relations defined in Section 8 of Schneiderman and Teichner's work. Pulling-apart conjecture. The components of $A$ can be pulled apart if and only if

$$\lambda_2(A)\in\Lambda_2/\operatorname{INT}_2(A)$$

is zero. This gives a proposed complete obstruction for pulling apart four components in the first partially understood non-repeating case. The conjecture remains open.