ag.algebraic geometry – Distinguishing ample divisors by minimally intersecting curves on a projective simplicial toric variety

My question has an easily formulated generalization, which I will state first. Let $sigma subseteq mathbf{R}^n$ be a strongly convex polyhedral cone. For each minimally generating lattice point $m in sigma^o cap mathbf{Z}^n$ of the interior cone $sigma^o subseteq sigma$, let $S(m) subseteq sigma^{vee} cap mathbf{Z}^n$ denote the set of lattice points $u$ with $langle u,m rangle = 1$. My question is:

Does $S(m) = S(m’)$ imply that $m = m’$?

As a special case, assume that $sigma$ is the nef cone of a simplicial projective toric variety $X_{Sigma}$. Then my question seems to amount to the following:

If $D_1$ and $D_2$ are two ample divisors minimally generating in the ample cone, then does $D_1 cdot C = 1 Leftrightarrow D_2 cdot C = 1$ for all effective curves $C$ imply that $D_1 = D_2$?

This is the case I am most interested in.