Cavenagh–Hämäläinen–Lefevre–Stones completion conjecture for partial multi-Latin squares

Let $r$ and $\lambda$ be positive integers. A simple partial $\lambda$-Latin square of order $r$ is an $r\times r$ array whose cells contain multisets of at most $\lambda$ symbols, with each symbol occurring at most $\lambda$ times in each row and column, and with no repeated symbol in any cell. A simple $\lambda$-Latin square of order $n$ is an $n\times n$ array in which each cell contains a multiset of $\lambda$ symbols, each symbol occurs at most $\lambda$ times in each row and column, and no cell contains repeated symbols.

Cavenagh–Hämäläinen–Lefevre–Stones' conjecture. For each $r,\lambda\in\mathbb N$, there exists an integer $n(r,\lambda)$ such that, for every $n\geq n(r,\lambda)$, every simple partial $r\times r$ $\lambda$-Latin square can be extended to a simple $n\times n$ $\lambda$-Latin square.

The conjecture asks for a uniform sufficiently large order guaranteeing completion of every simple partial multi-Latin square. The paper states that it confirms this conjecture, while the supplied status for this candidate is open; the status should therefore be checked against the paper's final completion result.