Infinitude conjecture for quartic Fermat near misses

Let $n\in\mathbb Z$ and consider the Diophantine equation

$$x^4+y^4-w^2=n.$$

Quartic near-miss infinitude conjecture. If this equation has an integer solution, then it has infinitely many integer solutions. The conjecture is suggested by the paper's non-thinness results for several families of values of $n$, while the general case remains open.