This came up in a discussion with a colleague of mine, who studies PDEs. He was asking for a function $f \colon \mathbb{N} \rightarrow \mathbb{N}$ such that, for all but finitely many $n$, the equation $$ x^4 + y^4 = n $$ has at most $f(n)$ solutions in integers $x,y$. My own feeling was that there should be a constant function $f$ that does the trick.

Indeed, if the "weak Lang conjecture" (see below) is true, then even the number of rational solutions to the above equation is bounded uniformly over all $n$. Assuming the weak Lang conjecture then, we can take $f$ to be a constant function. My question is whether we can prove this statement without assuming any unknown conjecture:

Does there exist $N \in \mathbb{N}$ such that, for all $n \in \mathbb{N}$, the equation $x^4+y^4=n$ has at most $N$ solutions in integers $x,y$?

The statement of the weak Lang conjecture is as follows: if $X$ is a variety of general type over a number field $K$, then the set $X(K)$ of rational points of $X$ is not Zariski dense. In their article "Uniformity of rational points" (JAMS, 1997), Caporaso, Harris, and Mazur prove that this conjecture implies the existence of constants $B(K,g)$ such that every smooth genus $g$ curve $X$ over a number field $K$ has at most $B(K,g)$ rational points (see Theorem 1.1 in their paper; note that they probably assume $X$ irreducible as well, although they do not explicitly state this).