Joint-distribution independence conjecture for consecutive values of $\Omega$

Let $\Omega(n)$ denote the total number of prime factors of $n$, counted with multiplicity, let $[N]=\{1,\ldots,N\}$, and for a finite set $S$ write $\mathbb{E}_{x\in S}f(x)=|S|^{-1}\sum_{x\in S}f(x)$. Joint-distribution independence conjecture. For any $k\in\mathbb{N}$ and any bounded function $a\colon\mathbb{N}^k\to\mathbb{C}$,

$$\mathbb{E}_{n\in[N]}a\bigl(\Omega(n),\Omega(n+1),\ldots,\Omega(n+k-1)\bigr)=\mathbb{E}_{(n_1,\ldots,n_k)\in[N]^k}a\bigl(\Omega(n_1),\ldots,\Omega(n_k)\bigr)+o_{N\to\infty}(1).$$

This is presented as an equivalent formulation of the almost-prime independence conjecture and expresses asymptotic independence of the joint distribution of consecutive values of $\Omega$. The paper states that the equivalent formulation remains open.