Nilsequence structure conjecture for Hardy-field multicorrelations

Let $f_1,\ldots,f_k$ be functions of polynomial growth from a Hardy field, let $W$ be a compatible weight, and let $(X,\mathcal{B},\mu,T)$ be an invertible measure-preserving system. For $h_0,\ldots,h_k\in L^\infty(X)$, define

$$\alpha(n)=\int_X h_0\cdot T^{\lfloor f_1(n)\rfloor}h_1\cdots T^{\lfloor f_k(n)\rfloor}h_k\,\mathrm{d}\mu.$$

Nilsequence structure conjecture. There exist a nilmanifold $Y=G/\Gamma$, a continuous function $F\in C(Y)$, a point $y\in Y$, elements $a_1,\ldots,a_k\in G$, and a sequence $\nu(n)$ such that

$$\alpha(n)=F\left(a_1^{\lfloor f_1(n)\rfloor}\cdots a_k^{\lfloor f_k(n)\rfloor}y\right)+\nu(n),$$

for $n\in\mathbb{N}$, and

$$\lim_{N\to\infty}\frac{1}{W(N)}\sum_{n=1}^Nw(n)|\nu(n)|=0.$$

This would give a nilmanifold model for multicorrelation sequences up to a negligible weighted error; the source states it as a conjecture and supplies no resolution.