Conjecture on eventual completion of simple partial multi-latin squares

Let $k$ and $m$ be positive integers. A simple partial $k$-latin square is a partial $k$-latin square in which no symbol is repeated within a cell. Eventual embedding conjecture. There exists a finite value $n(m,k)$ such that, for every $n\geq n(m,k)$, every simple partial $k$-latin square $P$ of order $m$ embeds in a simple $k$-latin square of order $n$. The preceding theorem establishes the analogous embedding result without the simplicity requirement; determining whether simplicity can also be preserved remains open.