Positive geometry conjecture for the nonnegative ABCT variety

Let $\operatorname{Gr}(2,n)_{>0}$ denote the positive part of the Grassmannian, let $\theta$ be the rational Veronese map, and define the ABCT variety by

$$V(k,n):=\overline{\theta(\operatorname{Gr}(2,n))}\subset\operatorname{Gr}(k,n).$$

Positive geometry conjecture. The closure

$$V(k,n)_{\geq 0}:=\overline{\theta(\operatorname{Gr}(2,n)_{>0})}$$

is a positive geometry.

The claim identifies the positive part of the Veronese image as a positive geometry, but the supplied text gives no result resolving it.