The complete bipartite graph size Ramsey number conjecture

Let $K_{s,t}$ denote the complete bipartite graph with parts of sizes $s$ and $t$, and let $hat r(H)$ denote the size Ramsey number of a graph $H$. For functions of the parameters, write $f=\Theta(g)$ when they are bounded above and below by positive constant multiples of $g$. Complete bipartite size Ramsey conjecture. For all $s\leq t$,

$$hat r(K_{s,t})=\Theta(s^2t2^s).$$

In particular,

$$hat r(K_{t,t})=\Theta(t^3 2^t).$$

The paper proves this order of magnitude when $t=\Omega(s\log s)$ and conjectures that it extends to the full range $s\leq t$.