The topological-surface asymptotic bondage-number conjecture

Let $G$ be a connected graph of orientable genus $h$ and non-orientable genus $k$. Let $b(G)$ be its bondage number and $\Delta(G)$ its maximum vertex degree. Let $c_h$ and $c'_k$ be constants depending, respectively, on the orientable and non-orientable genera of $G$, and let $o(h)$ and $o(k)$ denote the genus-dependent terms appearing in the conjecture. Topological-surface bondage-number conjecture.

$$b(G)\le\min\{c_h,\,c'_k,\,\Delta(G)+o(h),\,\Delta(G)+o(k)\}.$$

The conjecture proposes that refinements of the paper's surface-embedding arguments yield both constant and maximum-degree-based upper bounds across topological surfaces. The paper presents it as a general conjectural direction after proving explicit bounds, but no resolution is supplied.