Given a bipartite graph $G$ with two sides $A=\{a_1,\dots, a_n\}$ and $B=\{b_1,\dots, b_n\}$.
I define a ``pressing" operation $f_{a_i,a_j}(G)=G'$ for $i<j$: $G'[A\cup B\setminus\{a_i,a_j\}]=G[A\cup B\setminus \{a_i,a_j\}]$. $N_{G'}(a_i)=N_G(a_i)\cup N_G(a_j)$ and $N_{G'}(a_j)=N_G(a_i)\cap N_G(a_j)$.
It means for $b\in B$, if $\{b,a_j\}\in G$ and $\{b,a_i\}\not\in G$, then in $G'$, $\{a_i,b\}\in G'$ but $\{a_j,b\}\not\in G'$. It sounds like we push edges of $G$ to the left.
Similarly we define $f_{b_i,b_j}(G)=G'$ for $i<j$ by replacing $a$ by $b$.
I am wondering if is it true that after a finite sequence of pressing operations, we can obtain a bipartite graph $G'$ so that if $\{a_i,b_j\}\in G'$, then $\{ \{a_p,b_q\}\mid p\le i, q\le j \}\subseteq G'$?
This is a standard argument. We should prove two things. First:
Claim 1. We can't make arbitrarily many pressing operations that change the graph we have at every step.
Proof. For this, we want to define some monovariant that always changes predictably with every pressing operation. For example, the sum $$\sum_{i=1}^n i \cdot (\deg(a_i) + \deg(b_i))$$ will always decrease whenever a pressing operation $f_{a_i, a_j}$ or $f_{b_i, b_j}$ modifies the graph in some way. Since the sum is a positive integer for any graph, it can't decrease forever.
It follows that we can eventually reach a graph which cannot be changed by any pressing operation. So then we prove:
Claim 2. If a graph $G$ satisfies $f_{a_i, a_j}(G) = f_{b_i, b_j}(G) = G$ for all $i,j$ with $i<j$, then it has the property we want: for all $p,q,i,j$ with $p\le i$ and $q \le j$, if $a_i b_j \in E(G)$, then $a_p b_q \in E(G)$.
Proof. Check that whenever $p \le i$ and $q \le j$, and $a_i a_j \in E(G)$ but $a_p a_q \notin E(G)$, we either have $f_{a_p,a_i}(G) \ne G$ or $f_{b_q, b_j}(G) \ne G$. (It might be worthwhile to begin with the cases where either $p=i$ or $q=j$; they are slightly different but also slightly easier.)
Now we have a strategy for getting a graph of the type we want. All we have to do is choose pressing operations that actually do something. If we keep doing this for as long as we can, we eventually get a fixed point, and any fixed point is guaranteed to have the desired property.