The bipartite unique domination bound

Let $G=(V,E)$ be a finite simple bipartite graph without isolated vertices, with order $n=|V|$ and domination number $\gamma(G)=\gamma$. A unique minimum dominating set is a minimum dominating set of $G$ that is the only one. Define

$$\Phi(n,\gamma)=\max\left(0,n-3\gamma-2\left\lceil\frac{\gamma}{2}\right\rceil-\left\lfloor\frac{\gamma}{2}\right\rfloor+1\right).$$

Bipartite unique domination bound. If $G$ has a unique minimum dominating set, $\gamma\geq 2$, and $n\geq 3\gamma$, then its size $s(G)$ is bounded above by

m(n,\gamma)=2\gamma+2\left\lceil\frac{\gamma}{2}\right\rceil\left\lfloor\frac{\gamma}{2}\right\rfloor+\min\left\\{n-3\gamma,2\left\lceil\frac{\gamma}{2}\right\rceil-\left\lfloor\frac{\gamma}{2}\right\rfloor+1\right\\}\left(2\left\lceil\frac{\gamma}{2}\right\rceil+1\right)+\sum_{i=1}^{\Phi}\left(2\left\lceil\frac{\gamma}{2}\right\rceil+1+\left\lceil\frac{i}{2}\right\rceil\right).

This conjecture proposes the sharp edge bound for bipartite graphs with a unique minimum dominating set in the stated range, refining the general bounds for uniquely dominatable graphs and complementing known results for special cases such as domination number $2$.