Let $V$ be an $\mathbb{R}$-vector space of finite dimension, let $N$ be a $\mathbb{Z}$-structure on $V$, and let $M$ be its dual $\mathbb{Z}$-structure on the dual space $V^*$.

Let $\sigma\subseteq V$ be an $N$-rational pointed polyhedral cone and let $R$ be a ring. The intersection $\sigma^\vee_M$ of the dual cone of $\sigma$ with $M$ is a monoid, and thus gives rise to the $R$-algebra $R[\sigma^\vee_M]$. Its spectrum $X_\sigma(R)$ is known as *the affine toric scheme over $R$ associated with $\sigma$.*

Let $\tau\subseteq V$ be a further $N$-rational pointed polyhedral cone, and suppose that $\tau$ is a *subset* of $\sigma$. The above procedure then yields an injection of monoids $\sigma^\vee_M\hookrightarrow\tau^\vee_M$ and thus a morphism of $R$-schemes $h_{\tau,\sigma}(R)\colon X_\tau(R)\rightarrow X_\sigma(R)$.

It is a well-known and basic fact in the theory of toric varieties that if $\tau$ is a *face* of $\sigma$, then the above morphism of $R$-schemes $h_{\tau,\sigma}(R)$ is an open immersion. I wonder about the converse, and I would be surprised if the following is not true:

Conjecture:If the canonical morphism $h_{\tau,\sigma}(R)\colon X_\tau(R)\rightarrow X_\sigma(R)$ is an open immersionfor some non-zero ring $R$,then $\tau$ is a face of $\sigma$.

I tried to attack this in several ways but did not succeed so far. (One idea led to this more general question.) So, my question is whether this conjecture is known to be true, and of course about any source proving or disproving it.

(If this is of any use, note that by base change we can suppose that $R$ is nice (e.g., noetherian, a field, an algebraically closed field...), and that by faithfully flat descent we can even suppose that $R$ is a prime field.)