Conjectured removal of the logarithmic factor in tensor robust PCA recovery

Let $d$ be the tensor order, let $n_d$ denote the relevant tensor dimension, and consider the exact-recovery theorem for tensor robust PCA under the paper's stated assumptions. The assumption includes a factor $\ln^{\max\{2d-5,0\}}(n_d)$.

Tensor robust PCA recovery conjecture. The factor

$$\ln^{\max\{2d-5,0\}}(n_d)$$

can be removed from the assumption without invalidating the exact-recovery conclusion of the theorem.

This conjecture seeks sharper conditions for exact recovery of tensors of arbitrary order, extending the matrix-case perspective. The paper establishes the recovery theorem with the logarithmic factor present, while its removal remains open.