Higher-dimensional periodic Costas array conjecture

Let $A$ be an $m$-dimensional Costas array of order $n$ defined by a bijection

$$\varphi:[n_1] \times \cdots \times [n_k] \longrightarrow [n_{k+1}]\times\cdots\times [n_m],$$

where $k\geq m-k$. A Costas array is periodic Costas if its periodic extension has every corresponding window satisfying the Costas condition.

Higher-dimensional periodic Costas array conjecture. If $A$ is periodic Costas, then $n=2^k$. In particular,

$$n_1=n_2=\cdots=n_k=2.$$

This is proposed as a higher-dimensional analogue of the periodicity theorem for two-dimensional Costas arrays. The paper reports exhaustive computations supporting it, but no proof or resolution is given, so the conjecture remains open.