I'm concerned with the validity problem for sentences of first-order logic over finite words, i.e. $FO[\le]$ interpreted over finite subsets of $\mathbb{N}$. AFAIK it should be nonelementary.
However, I'm looking at the complexity of the levels of the alternation hierarchy, i.e., $\Sigma_n$ and $\Pi_n$ fragments of $FO[\le]$. For example, the satisfiability problem for Bernays-Schönfinkel formulae, those of the form $\exists^*\forall^*\phi$, a.k.a. $\Sigma_2$-formulae, is in general $\mathsf{NEXPTIME}$-complete, and this should hold also on words, is this correct? But then what is the complexity of satisfiability/validity for $\Sigma_n$-formulae for a fixed $n$?
I've found a lot of papers and surveys about the expressibility problem for these fragments, that is, to decide whether a given language can be expressed in a given fragment, but nothing on the computational complexity of the validity/satisfiability problem. I'm feeling like I'm missing something very trivial or commonly known.
Can you give me any reference?