The claim does not hold. A counterexample is given by $n=14$, $p=134123250258009499$ and correspondingly
$$N = 2197475332227227631617 = 193 \cdot 12289 \cdot 926510094425921.$$
It can be easily verified that
$$3^{(N-1)/2} \equiv 1 \pmod{N},$$
but $N$ is not prime.

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A couple more values of $N$ giving counterexamples:
$$300334937065845770469377,\ 80203520301265852381167617.$$

**ADDED.** Here is a list of [659 of counterexamples](https://gist.github.com/maxale/b09e4ac7889dccb1706db9bff17dbc3b) that I found.