Independence conjecture for almost-prime patterns

Let $\mathbb{P}_\ell$ denote the set of integers with the relevant almost-prime parameter $\ell$, let $\1_{\mathbb{P}_\ell}$ be its indicator, and let $\overline{\pi}_\ell(N)$ denote its normalized counting function on $[N]$. Write $[N]=\{1,\ldots,N\}$ and $\mathbb{E}_{n\in[N]}f(n)=N^{-1}\sum_{n=1}^Nf(n)$. Independence conjecture for almost primes. For all $k\in\mathbb{N}$,

$$\sum_{\ell_1,\ldots,\ell_k=1}^{\infty}\left|\mathbb{E}_{n\in[N]}\1_{\mathbb{P}_{\ell_1}}(n)\1_{\mathbb{P}_{\ell_2}}(n+1)\cdots\1_{\mathbb{P}_{\ell_k}}(n+k-1)-\overline{\pi}_{\ell_1}(N)\cdots\overline{\pi}_{\ell_k}(N)\right|=o_{N\to\infty}(1).$$

This conjecture would imply the expected asymptotic independence for almost-prime patterns for almost all parameters in the typical range, and the paper notes that it implies the functional Chowla conjecture. Its general validity remains open.