Classical randomized simulation conjecture for divergence-free quadratic ODEs

Let $\beta\geq 5/3$ and define

$$J=\sum_{i,j,k=1}^N|c_{ijk}|=O(\operatorname{polylog}(N)).$$

Assume there is an efficient procedure for sampling from the probability distribution induced by the normalized tensor $c/J$. Consider the quadratic ODE system

$$\frac{dX_i}{dt}=f_i(X)=\sum_{i,j,k=1}^N c_{ijk}X_jX_k,$$

with divergence-free drift $c$ and sparse initial condition $X(0)=X_0$. Classical simulation conjecture. There exists a classical randomized time-evolution algorithm that solves this system directly in ODE state space with polynomial runtime. The claim concerns a proposed classical counterpart to the quantum simulation method; its validity is left open in the supplied text.