Infinite-dimensional limit of the finite Riccati matrix

Let $\Phi^N(t)$ be the finite-dimensional matrix-valued function defined by the Riccati system in the source, with entries $\Phi^{N,i,j}(t)$. Consider the associated functions $\phi^k$ defined by the cited infinite-player system of ordinary differential equations. Infinite-dimensional Riccati-limit conjecture. The limit of each element in $\Phi^{N}(\cdot)$ exists as $N\to\infty$; equivalently, $\Phi^N(t)\to\Phi^\infty(t)$, where $\Phi^\infty(t)=(\Phi^{\infty,i,j}(t))_{i,j\in\mathbb N}$ is an infinite-dimensional lower-triangular matrix-valued function satisfying

$$\Phi^{\infty,i,j}(\cdot)\equiv 0 \quad\text{if } i<j,$$

and

$$\Phi^{\infty,i,j}(\cdot)\equiv \phi^{i-j} \quad\text{if } i\ge j,$$

with the functions $\phi^k$ given by the system of ordinary differential equations cited in the source. Establishing this convergence remains an open problem.