The bipartite linkless-embeddability conjecture

Let $G=(A\uplus B,E)$ be a bipartite linklessly embeddable graph, and let $<$ be a $(3,3)$-admissible order. Write $G^b$ for the balanced shift of $G$. Bipartite linkless-embeddability conjecture. If $|A|,|B|\geq 4$, then $K_{4,4}^-$ is not a subgraph of $G^b$, and consequently

$$E\leq 3|V(G)|-10.$$

In particular, all bipartite linklessly embeddable graphs are $(3,3)$-stress free; hence, if $A$ and $B$ each have size at least $3$,

$$E\leq 3|V(G)|-9.$$

This is motivated by the non-linkless embeddability of $K_{4,4}$ minus an edge. The source explicitly states that even the inequality $E\leq 3|V(G)|-9$ is open.