The density domination exponent conjecture for complete bipartite graphs

Let $a_1\ge a_2$ and $b_1\ge b_2$ be positive integers satisfying

$$a_2\ge b_2>1,\qquad a_1\le b_1,\qquad b_1+b_2\ge a_1+a_2.$$

For graphs $G$ and $H$, let $\rho(G,H)$ denote the density domination exponent. In particular, write $K_{a_1,a_2}$ and $K_{b_1,b_2}$ for complete bipartite graphs.

The density domination exponent conjecture.

$$\rho(K_{a_1,a_2},K_{b_1,b_2})=\max\left\{\frac{b_1b_2}{a_1a_2},\frac{b_1+b_2-1}{a_1+a_2-1}\right\}.$$

This conjecture gives the missing case in the paper's determination of density domination exponents for complete bipartite graphs. The supplied text does not state whether it has been resolved.