What can I say about the column space of the product of two nonnegative softmax matrices?

Question

Let $\mathcal S(\cdot)$ denote applying softmax transform along the first dimension. Given a nonnegative matrix $\mathcal S(\mathbf A \mathbf C^\top) \in \mathbb R_+^{m \times n}$ with variable matrices $\mathbf A$ and $\mathbf C$, convert it to the product of two nonnegative matrices with constant $\mathbf B$: $\mathcal S(\mathbf A \mathbf B^\top) \mathcal S(\mathbf B \mathbf C^\top) \in \mathbb R_+^{m \times n}$. All these matrices are at most of rank $d$: $\mathbf A \mathbf C^\top$, $\mathbf A \mathbf B^\top$, and $\mathbf B \mathbf C^\top$, where $d < \max(m,n)$. Denote $\operatorname{col}(\cdot)$ the column space of a matrix. I know that $\operatorname{col}(\mathbf A \mathbf B^\top) \subseteq \operatorname{col}(\mathbf A)$. Question 1: Is it equally true that $\operatorname{col}(\mathcal S(\mathbf A \mathbf B^\top) \mathcal S(\mathbf B \mathbf C^\top)) \subseteq \operatorname{col}(\mathcal S(\mathbf A \mathbf C^\top))$? Question 2: If yes, what can I say about the column space of the former matrix, namely $\mathcal S(\mathbf A \mathbf B^\top) \mathcal S(\mathbf B \mathbf C^\top)$, in terms of the constant matrix $\mathbf B$? Thank you so much!