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The answer to this question for any $n\geq 4$ is positive iff it is positive for $n=4$. If there is a Schoenfiles PL embedding $f\colon S^3\hookrightarrow S^4$, which is not PL equivalent to the equatorial inclusion, then $\Sigma^{n-4}f\colon S^{n-1}\hookrightarrow S^n$ is not PL locally flat, $n\geq 5$, by induction looking at the map of links $S^{n-2}=L_a(S^{n-1})\hookrightarrow L_{\Sigma^{n-3}f(a)}(S^{n})=S^{n-1}$. (Note that it would still be topologically locally flat, since $f$ is trivial topologically.) Otherwise, any such embedding is PL locally flat by induction, and therefore is trivial, see page 47 in "Introduction to piecewise-linear topology" by Rourke-Sanderson.