Let $a_1,\ldots,a_r,b_1,\ldots,b_r\in \mathbb{N}$ such that $\sum_{i}\frac{1}{a_i}>1$. For any finite sequence of $n$ (not necessarily distinct) integers $A=(x_1,\ldots,x_n)$ let $T(A)$ denote the sequence of length $rn$ given by $$(a_ix_j+b_i)_{1\leq j\leq n, 1\leq i\leq r}.$$ Prove that, if $A_1=(1)$ and $A_{i+1}=T(A_i)$, then there must be some $A_k$ with repeated elements.

Let $a_1,\ldots,a_r,b_1,\ldots,b_r\in \mathbb{N}$ such that $\sum_{i}\frac{1}{a_i}>1$. For any finite sequence of $n$ (not necessarily distinct) integers $A=(x_1,\ldots,x_n)$ let $T(A)$ denote the sequence of length $rn$ given by $$(a_ix_j+b_i)_{1\leq j\leq n, 1\leq i\leq r}.$$ Prove that, if $A_1=(1)$ and $A_{i+1}=T(A_i)$, then there must be some $A_k$ with repeated elements.

This is true. This appears to have first been shown by Klarner [Kl82], with a generalisation given by Kolpakov and Talambutsa [KoTa22]. Essentially the same proof was found independently by Barreto in the comment section.