Multiple ergodic convergence for squared fractional powers

Let $(X,\mathcal{B},\mu,T)$ be a measure-preserving system, let $c_1,\ldots,c_k$ be distinct positive non-integers, and let $f_1,\ldots,f_k\in L^{\infty}(\mu)$. The multiple ergodic convergence problem. Do the averages

$$\frac{1}{N}\sum_{n=1}^{N} T^{\lfloor n^{c_1}\rfloor^2}f_1\cdot\ldots\cdot T^{\lfloor n^{c_k}\rfloor^2}f_k$$

converge in mean? This is non-trivial even when the fractional powers are replaced by general non-integer real polynomials; the case $k=1$ has a positive answer, while the general multiple case remains open.