Optimal computational error rate for moment and cumulant tensor estimation

Let $p$ be the dimension, $n$ the sample size, and $d$ the tensor order. Consider estimating $d$-th order moment or cumulant tensors in spectral norm, in the intermediate regime

$$p^{d/2} \ll n \ll p^{d-1}.$$

Optimal estimation rate conjecture. The computationally optimal error rate is

$$\frac{p^{d/2}}{n}.$$

This rate is strictly worse than the minimax statistical rate $\sqrt{p/n}$ in the stated regime. Whether efficient procedures can improve on the sample moment or cumulant tensor rate remains open, particularly without additional structural assumptions such as low rank.