HMMT 2023 T6

For any odd positive integer $n$, let $r(n)$ be the odd positive integer such that the binary representation of $r(n)$ is the binary representation of $n$ written backwards. For example, $r(2023)=$ $r\left(11111100111_{2}\right)=11100111111_{2}=1855$. Determine, with proof, whether there exists a strictly increasing eight-term arithmetic progression $a_{1}, \ldots, a_{8}$ of odd positive integers such that $r\left(a_{1}\right), \ldots$, $r\left(a_{8}\right)$ is an arithmetic progression in that order.