The order conjecture for entire quasiregular Yosida mappings

Let $n\geq 2$, and let $f:\mathbb{R}^n\to\mathbb{R}^n$ be an entire quasiregular Yosida mapping, meaning that the family of translates

$$\{f(x+a):a\in\mathbb{R}^n\}$$

is normal. Order conjecture. The order of growth of $f$ is at most $n-1$.

For quasiregular Yosida mappings into $S^n$, the paper proves the upper bound $n$. The conjectured improvement to $n-1$ is motivated by the corresponding theorem for entire holomorphic Yosida functions in the plane, but remains open for entire quasiregular mappings.