Is "decomposition" of every Friedlander-Iwaniec prime unique?

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If by Friedlander-Iwaniec primes we mean primes of form $a^2 + b^4$, can we be sure that $a$ and $b$ here are always unique?

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$$97 = 4^2 + 3^4 = 9^2 + 2^4$$

But that sort of thing is the only problem, since the representation of a prime as the sum of two squares (if it exists, i.e. $p = 2$ or $p \equiv 1 \pmod{4}$) is unique.