The sets are defined in $\mathbb{R}_+^n$ $(n\geq 1)$. The relative interior of a convex set $C$ is denoted $\mathring C$. The codimension of a submanifold $M$ of a manifold $N$ is denoted codim$(M|N)$.
Let $S$ be the $n$-simplex:
$$S=\left\{x\in\mathbb{R}_+^n,\,\sum_{i=1}^n x_i=1\right\}$$
and $E$ be a linear subspace such that $E\cap\mathring S\neq\varnothing$ (hence, $E$ contains a vector with positive coordinates) and codim$(E|\mathbb{R}^n)=2$. For $F$ a face of $E\cap S$, one can see that there exists a unique face $G$ of $S$ such that $\mathring F\subset\mathring G$. Hence, $E\cap G=F$.
Question: Assuming codim$(\mathring F|\mathring G)=2$, if a face $G'$ of $S$ satisfies $G'\cap E=F$, do $G'=G$?