Extremal edge conjecture for unbalanced bipartite 1-planar graphs

Let $x$ and $y$ be integers with $x\geq 3$ and $y\geq 6x-12$, and let $G$ be a bipartite 1-planar graph whose partite sets have sizes $x$ and $y$. Extremal edge conjecture. The graph $G$ has at most

$$2|V(G)|+4x-12$$

edges. This conjecture proposes the optimal upper bound in the sufficiently unbalanced regime, where the smaller part has size at most roughly one sixth of the larger part. The paper's preceding results provide matching constructions and bounds in this setting, but the stated optimality remains open.