Computational hardness of consistent moment and cumulant tensor estimation

Let $p$ be the dimension, $n$ the sample size, and $d$ the tensor order. Consider estimating the order-$d$ moment or cumulant tensors of a sub-Gaussian random vector, with consistency measured in tensor spectral norm. Consistent estimation hardness conjecture. When

$$n \ll p^{d/2},$$

there is no efficient algorithm that can consistently estimate the order-$d$ cumulant or moment tensors of any sub-Gaussian random vector. The conjecture is motivated by the matching computational threshold for hypothesis testing and by the fact that the spectral-norm loss is NP-hard to compute or approximate; it remains open.