The fan embedding conjecture for complete bipartite graphs

Let $K_{m,n}$ be a complete bipartite graph with independent vertex sets $\{a_1,\dots,a_m\}$ and $\{b_1,\dots,b_n\}$. Place $a_i$ at $(i,0)$ on the $x$-axis and $b_j$ at $(0,j)$ on the $y$-axis, and join each $a_i$ to each $b_j$ by the corresponding straight line segment; this is the fan embedding of $K_{m,n}$. Here $mnl(G)$ denotes the minimum number of links among spatial embeddings of the graph $G$. Fan embedding conjecture. The fan embedding of $K_{m,n}$ realizes

$$mnl(K_{m,n}).$$

The conjecture is motivated by the fact that the fan embedding realizes the minimum for $K_{4,n}$, where the minimum is $2\binom{n}{4}$. Whether it is minimal for all complete bipartite graphs remains open.