Before revision:  However, we also exhibit some sufficient conditions on A and \mathcal{S}, under which a related set \mathcal{X} := \text{Span}{\{x_{b,\omega} - x_{b,\mu} \mid b \in \mathbb{F}^n, \omega,\mu > -\lambda_{min}(A)\}} has dimension equal to \mathbf{Ind}_A(\mathcal{S}).
After revision:  However, we also exhibit some sufficient conditions on A and \mathcal{S}, under which \mathcal{X} := \text{Span}{\{x_{b,\omega} - x_{b,\mu} \mid b \in \mathbb{F}^n, \omega,\mu > -\lambda_{min}(A)\}} has dimension equal to \mathbf{Ind}_A(\mathcal{S}).