Existence of a sequence with prescribed limit and satisfying a certain inequality II

This is a more specific variation of the question in the post

Existence of a sequence with prescribed limit and satisfying a certain inequality

Suppose you have two infinite sequences $\{a_n\},\{b_n\}$, with $0<a_n<b_n < 1$ for each $n$, such that both $a_n, b_n \to 1$ as $n \to \infty$. Further assume that $a_n/b_n \to 1$ as $n \to \infty$.

Does there exist a sequence $\{s_n\}$ with $s_n \to 1$ as $n \to \infty$ such that
$$
(1 – a_n) \leq (1-b_n)s_n
$$ for $n$ large enough?

The counterexamples given in the post above used sequences $a_n, b_n$ which grew asymptotically different (but which had the same limit). This question forces that $a_n \sim b_n$.

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