The non-principal pair of posets conjecture for the incidence graph of $K_4$

Let $G_4=(A,B,E)$ be the bipartite incidence graph of the complete graph $K_4$, with $A=E(K_4)$, $B=V(K_4)$, and $(e,v)\in E$ exactly when $v\in e$. Let $P_{G_4,A}$ and $P_{G_4,B}$ be the associated posets, and let $\mathop{}\!\mathrm{La}(n,\{P_{G_4,A},P_{G_4,B}\})$ denote the maximum size of a family in $2^{[n]}$ containing neither poset. The non-principal pair of posets conjecture.

$$\mathop{}\!\mathrm{La}(n,P_{G_4,A})-\mathop{}\!\mathrm{La}(n,\{P_{G_4,A},P_{G_4,B}\})=\Omega\left(\binom{n}{n/2}\right).$$

This predicts that the family of posets $\{P_{G_4,A},P_{G_4,B}\}$ is not asymptotically principal: forbidding both posets costs a positive-order middle-layer term beyond forbidding $P_{G_4,A}$ alone. The source gives no resolution, so the conjecture remains open.