Generalized orthogonality of polynomial correlation sequences

Let $k\in\mathbb{N}$, let $(X,\mu,T)$ be an invertible measure-preserving system, let $P=\{p_1,\ldots,p_r\}$ be a set of essentially distinct integral polynomials, let $h$ be an integral polynomial with $h\notin \textup{\textsf{{WP}}}(P)$, and let $\alpha\in\mathbb{R}\setminus\mathbb{Q}$. If $g:\mathbb{T}\to\mathbb{C}$ is Riemann integrable and $f_0,f_1,\ldots,f_r\in L^\infty(\mu)$, then

Generalized orthogonality conjecture.

$$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N\left(g(h(n)\alpha)-\int_{\mathbb{T}}g\,d\lambda\right)\int f_0T^{p_1(n)}f_1\cdots T^{p_r(n)}f_r\,d\mu=0.$$

The conjecture seeks to extend the Weyl-system orthogonality result to arbitrary invertible measure-preserving systems; the source notes technical complications in lifting the result to nilsystems. The source does not state whether it has been resolved.