Suppose I have two lists $\alpha_1,\dots,\alpha_k$ and $\beta_1,\dots,\beta_k$ of real numbers such that all $2k$ numbers are mutually algebraically-independent over the rationals. For each $i \in \{1,\dots,k\}$, let $\phi_k$ be the affine linear map $x \mapsto \alpha_i x + \beta_i$.

Let $I$ be the image of the integers, $\mathbb Z$, under some finite sequence of applications of these maps and/or their inverses. (That is, $I = \phi_{i_\ell}^{e_\ell} \circ \cdots \circ \phi_{i_1}^{e_1} ({\mathbb Z})$ for some $\ell \geq 1$, $e_j \in \{+1,-1\}$, and $i_j \in \{1,\dots,k\}$).

Is the sequence of maps $\phi_{i_1}^{e_1}, \dots, \phi_{i_\ell}^{e_\ell}$ is reconstructible from $I$? (Is there a unique sequence of applications of these maps and inverses such to find $I$ from ${\mathbb Z}$?)

It seems that a formal statement would be that these maps generate a free group within the group of affine linear maps. My intuition is that two distinct ways to generate an element of $I$ with these maps would create an algebraic relation between these algebraically-independent elements, but explicitly building the relation is cumbersome.