Let $\rho$ be a state (positive trace class operator with trace equal to $1$) on $H\otimes H$, where $H$ is a separable Hilbert space (not necessarily finite dimensional). We say that $\rho$ is countable separable if $\rho=\sum_{i=1}^{\infty}\lambda_i\alpha_i\otimes\beta_i $, where $\alpha_i,\beta_i$ are states on $H$ and $\lambda_i\geq 0$ such that $\sum_{i=1}^{\infty}\lambda_i=1$.
Let $\{e_n\}$ be an orthonormal basis for $H$ and let $P_n$ be the orthogonal projection on to the space of span $\{e_i:1\leq i\leq n\}$. Are the following two statements equivalent?

1. $P_n\rho P_n$ is countable separable, for all $n$.
2. $\rho$ is countable separable.

Clearly, $1 \implies 2$. What about the converse?