Complex relative entropy principle for imaginary arguments

Let $E_N(z;\sigma)$ be the expected-value polynomial defined for the multivariate normal random polynomial model, and let $y\in\mathbb{R}$. Consider positive and negative subsequences of $\{E_N^{1/N}(iy;\sigma)\}_{N\in\mathbb{N}}$. The complex continuum free-energy functional $\mathfrak{h}_{y;\sigma}(\dot\varsigma)$ is defined by allowing normalized complex densities relative to the signed measure $\nu_{iy;\sigma}$, with finite second moment.

Complex relative entropy principle. For each positive subsequence, the limit points of the normalized critical complex entropies are contained in the real parts of the critical values of $\mathfrak{h}_{y;\sigma}$, while for each negative subsequence they are contained in the critical values themselves:

$$\operatorname{limpt}\,\frac{1}{N'}\operatorname{crit}_{\varsigma^{(N')}}{\cal H}_{y;\sigma}^{(N')} (\dot\varsigma^{(N')})\subset \operatorname{Re}\mathfrak{e}\left(\operatorname{crit}_{\varsigma}\,\mathfrak{h}_{y;\sigma}(\dot\varsigma)\right)$$

and

$$\operatorname{limpt}\,\frac{1}{N'}\operatorname{crit}_{\varsigma^{(N')}}{\cal H}_{y;\sigma}^{(N')} (\dot\varsigma^{(N')})\subset \operatorname{crit}_{\varsigma}\,\mathfrak{h}_{y;\sigma}(\dot\varsigma).$$

This heuristic principle is proposed to explain the possible large-$N$ limit curves for imaginary $z$, including sign changes that are not obtained by direct analytic continuation of the real-$z$ formulas. Its status is conjectural and the source presents heuristic and graphical rather than rigorous evidence.