Optimal correlational bound in the highly correlated regime

Let $\Phi\in\mathfrak{h}\wedge\mathfrak{h}$ be normalized with canonical form

$$\Phi=\sum_{k=1}^{\infty}\lambda_{k}u_{k}\wedge v_{k}.$$

Write $\lambda_{\max}=\sup_{k}\lambda_k$. Highly correlated regime conjecture. There exists a constant $C>0$, independent of $\Phi$, such that for every $N\in2\mathbb{N}$ with $N\lambda_{\max}^{2}\leq1$,

$$\sup_{\Psi\in\bigwedge^{N}\mathfrak{h},\,\left\Vert \Psi\right\Vert =1}\left\langle \Phi,\gamma_{2}^{\Psi}\Phi\right\rangle \leq N\left(1-\frac{N-2}{2}\sum_{k=1}^{\infty}\lambda_{k}^{4}+C(N\lambda_{\max}^{2})^{2}\right).$$

The conjecture concerns the regime in which $N\lambda_{\max}^{2}$ is small and asserts that the asymptotics of the lower bound are optimal. The stronger bound without the error term is shown in the surrounding discussion to fail for general $\Phi$, but this counterexample does not apply in the highly correlated regime; the stated asymptotic upper bound remains open.