The Parisi PDE conjecture for the enriched bipartite free energy

Let $\mathcal Q$ denote the space of order parameters for the ultrametric external fields, let $q=(q_1,q_2)\in\mathcal Q^2$, and let $F_N:\mathbb{R}_+\times\mathcal Q^2\to\mathbb{R}$ be the enriched free energy of the bipartite model. Set $\psi:=F_1(0,\cdot)$.

Parisi PDE conjecture. The enriched free energy $F_N$ should converge to a function $f:\mathbb{R}_+\times\mathcal Q^2\to\mathbb{R}$ solving

$$\begin{cases} \partial_t f-\int_0^1\partial_{q_1}f\,\partial_{q_2}f=0 & \text{on }\mathbb{R}_+\times\mathcal Q^2,\nf(0,\cdot)=\psi & \text{on }\mathcal Q^2. \end{cases}$$

This is the proposed Hamilton–Jacobi or Parisi-type description of the limiting free energy for the bipartite model, extending the analogous construction for the Sherrington–Kirkpatrick model. The statement is presented as an expectation, and no resolution is supplied here.