My argument is not the most sound one, I would call more of a proof idea. My idea is a variation of Ladner's theorem, where instead of padding we use alternations of quantifers.
All levels of $PH$ have complete problems with a constant number of quantifier alternations. Consider now quantified formulas with at most $f(n)$ alternations where $f(n)\in o(1)\cap \omega(n)$, i.e. a sublinear non-constant function. For every such $f(n)$, let $f(n)-SAT$ be the language of satisfiable quantified formulas with at most $f(n)$ alternations.
It is apparent that $f(n)-SAT$ is in $PSPACE$ as it is a special version of $TQBF$, however I believe it is not in $PH$ and it is not $PSPACE$-complete.
Suppose that for some constant $f(n)$, $f(n)-SAT$ is $PSPACE$-complete. Then by definition,TQBF is Karp-reducible to $f(n)-SAT$. Thus we have a reduction that can limit the number of alterations from at most $n$ to at most $f(n)$. The same reduction, perhaps applied repetitively, could be probably used to reduce $f(n)$ to a constant number of alternations, thus it would be in $PH$. That would lead to a contradiction of the hypothesis that $PH \neq PSPACE$.
Therefore, $f(n)-SAT$ canoot be $PSPACE$-complete and since such a reduction does not exist, neither in $PH$.
One weak point of this idea is that perhaps the reduction can reduce the alternations to a small function, that is however in $o(1)$.