Let $(X,L)$ be a polarized toric $K$-variety. Pick $x_{1},...,x_{N}\in N_{\mathbf{Q}}$ and set $\mu_{w}:=\sum_{i}w_{i}\delta_{j(x_{i})}$ for each $w\in\mathbf{R}_{+}^{N}$. Then for a dense set of $w\in\mathbf{R}_{+}^{N}\cap\{\sum_{i}w_{i}=\deg L\}$ the semipositive toric metric $\|\cdot\|$ solving

$$c_{1}(L,\|\cdot\|_{w})^{n}=\sum_{i}w_{i}\delta_{j(x_{i})}.$$

is a model metric.