The few-symbol completion condition, including order zero

Write $[n]=\{0,\ldots,n-1\}$ for $n\in\mathbb N$, with $[0]=\varnothing$. A partial array of order $n$ is a map $P:[n]^2\to[n]\cup\{\bot\}$, with $\bot$ denoting an empty cell. Write $F(P)=\{(i,j):P(i,j)\ne\bot\}$ and $U(P)=\{a\in[n]:\exists i,j,\ P(i,j)=a\}$. It is partial Latin when no symbol repeats within a row or column. A completion is a map $L:[n]^2\to[n]$ injective in each row and column, with $L(i,j)=a$ whenever $P(i,j)=a\ne\bot$. For every $n\in\mathbb N$ and partial Latin $P$ of order $n$,

imply that $P$ has a completion of order $n$.