Nonexistence of Whitney towers with the framed Arf tree invariant

Let $A:S^2\cup S^2\looparrowright B^4$ support a Whitney tower $\mathcal W$, and let $t(\mathcal W)$ denote its associated tree invariant. Let $T$ be the sum of trees in Figure 8 of the source, possibly together with higher-order trees. Framed Arf tree restriction. There does not exist such an $A$ supporting $\mathcal W$ with

$$t(\mathcal W)=T.$$

This is another formulation of the conjecture that the Bing double of any knot with non-trivial classical Arf invariant does not bound an order-six framed Whitney tower. The claim remains open.