The group of quotients of the semigroup of invertible nonnegative matrices over local rings

In this paper, we prove that for a linearly ordered local ring $R$ with $1/2$ the group of quotients of the semigroup of invertible nonnegative matrices $\mathrm G_n(R)$ for $n \geq 3$ coincides with the group $\mathrm{GL}_n(R)$.