Quadratic universal graph conjecture for bipartite permutation graphs

For each positive integer $n$, let $U_n$ be a bipartite permutation graph containing every bipartite permutation graph on $n$ vertices as an induced subgraph, and let $u(n)$ be the minimum possible number of vertices of such a universal graph.

Quadratic universal graph conjecture. The minimum number of vertices in a bipartite permutation graph containing all $n$-vertex bipartite permutation graphs is

$$\Omega(n^2).$$

This conjecture asserts that the optimal universal bipartite permutation graph has quadratic size, matching the intended order of the known construction.