Maximality conjecture for the graph families $XH^k$ and $YH^k$

For each integer $k\geq 1$, let $XH^k$ and $YH^k$ denote the graph families constructed in the paper. Maximality conjecture for $XH^k$ and $YH^k$. The graphs $XH^k$ and $YH^k$ are maximal 1-planar graphs for all $k\geq 1$. The paper proves this only for $k=1$, so the assertion for general $k$ remains open.