Does every $n\times n\times n$ Latin cube contain a Latin transversal?

In 1967 H. J. Ryser conjectured that every Latin square of odd order has a Latin transversal. Similar to Latin squares, we may consider Latin cubes.

QUESTION: Let $n$ be any positive integer. Does every $n\times n\times n$ Latin cube contain a Latin transversal?

Let $N$ be any positive integer. In 2008, I proved that for the $N\times N\times N$ Latin cube over $\mathbb Z/N\mathbb Z$ formed by the Cayley addition table, each $n\times n\times n$ subcube with $n\le N$ contains a Latin transversal (cf. my paper An additive theorem and restricted sumsets). Motivated by this, in the 2008 paper I conjectured that my above question has a positive answer.

Any comments are welcome!

• Is an $n\times n\times n$ subcube essentially $J_1\times J_2 \times J_3$ where $J_i\subset \mathbb{Z}/N\mathbb{Z}$ for $i=1,2,3,$ and $|J_i|=n$? – kodlu Jun 24 '18 at 23:41
• @kodlu Yes, your understanding is correct. – Zhi-Wei Sun Jun 25 '18 at 16:47