The strong saturation conjecture for bipartite graphs

Let $S_n(p,q)$ be the minimum number of edges in an $n\times n$ bipartite graph such that adding any new edge between its two classes creates a copy of $K_{p,q}$. Let $\overrightarrow{S}_{n}(p,q)$ denote the directed analogue, where the copies of $K_{p,q}$ have their $p$ vertices in the first class and their $q$ vertices in the second class. Strong saturation conjecture. For every $p,q$ there is an integer $n_0$ such that for every $n\ge n_0$,

$$S_n(p,q)=\overrightarrow{S}_{n}(p,q)-\left\lfloor\frac{(q-p)^2}{4}\right\rfloor.$$

The directed quantity is known to equal $(p+q-2)n-(p-1)(q-1)$, while the displayed construction gives the stated upper bound for the undirected quantity. The conjecture asserts that this bound is best possible for sufficiently large $n$, so the gain from allowing undirected copies is an additive constant.