Conjecture on polynomial-time classical simulation with strong dissipation

Let $u_0$ be an observable depending only on the first $k$ coordinates, so that

$$u_0(x_1,\dots,x_N)=u_0(x_1,\dots,x_k,0,\dots,0).$$

Let $\lambda_i$ be dissipation rates satisfying $\lambda_i\propto i^p$ for an integer $p\in\{1,2,\dots\}$. Let $v(t,x)$ denote the expected value of the observable at time $t$. Classical Euler--Maruyama conjecture. There exists a classical Euler--Maruyama-type numerical algorithm that estimates $v(t,x)$ with runtime polynomial in $k$ and $\epsilon^{-1}$. The conjecture is motivated by the reduced effective dimension under increasing dissipation rates, but the supplied text does not establish such an algorithm.