Arratia's divisibility conjecture for prime factors of a uniform random integer

Let $N(n)$ be uniformly distributed on $\{1,\ldots,n\}$, with prime factorization

$$N(n)=\prod_{p\le n}p^{C_p(n)}.$$

For each prime $p\le n$, let $Z_p$ be an independent geometric random variable with parameter $\frac{1}{p}$ and range $\mathbb{Z}_{\ge 0}$, and define

$$M(n)=\prod_{p\le n}p^{Z_p}.$$

A coupling of $M(n)$ and $N(n)$ is a joint construction preserving their respective distributions. Arratia's conjecture. For all $n\ge 1$, it is possible to construct $N(n)$, $M(n)$, and a prime $P(n)$ such that

$$N(n)\mid M(n)P(n)$$

always. Equivalently, there exists a coupling such that

$$\sum_{p\le n}\left(C_p(n)-Z_p\right)^+\le 1$$

always. The conjecture proposes an especially strong coupling between the dependent prime-factor exponent process of a uniform random integer and its independent geometric limit process; the supplied text does not state whether it has been resolved.