Totally ergodic convergence conjecture for power functions

Let $f_1,\ldots,f_k$ be linearly independent functions of the form

$$f_i(t)=a_1t^{c_1}+\cdots+a_dt^{c_d},\qquad a_i,c_i\in\mathbb{R},\quad c_i>0.$$

Let $(X,\mathcal{B},\mu,T)$ be a totally ergodic measure-preserving system and let $h_1,\ldots,h_k\in L^\infty(X)$. Power-function convergence conjecture. Then

$$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N T^{\lfloor f_1(n)\rfloor}h_1\cdots T^{\lfloor f_k(n)\rfloor}h_k =\prod_{i=1}^k\int_Xh_i\,\mathrm{d}\mu$$

in $L^2(X)$. This interpolates between the known integer-polynomial totally ergodic case and the known all-noninteger-power case; the general mixed-power assertion is posed as an open conjecture.