Extension of endomorphisms of the subsemigroup $\mathrm{GE}^+_2(R)$ to endomorphisms of $\mathrm{GE}^+_2(R[x])$, where $R$ is a partially-ordered commutative ring without zero divisors

Let $R$ be a partially ordered commutative ring without zero divisors, $G_n(R)$ be the subsemigroup of $\mathrm{GL}_n(R)$ consisting of matrices with nonnegative elements, and $\mathrm{GE}^+_n(R)$ be its subsemigroup generated by elementary transformation matrices, diagonal matrices, and permutation matrices. In this paper, we describe in which cases endomorphisms of $\mathrm{GE}^+_2(R)$ can be extended to endomorphisms of $\mathrm{GE}^+_2(R[x])$.