Non-trivial solutions to $x^4 - Ny^4 = 1$ where $N = a^4 + b^4 + c^4$

I am investigating the Diophantine equation $$x^4 - Ny^4 = 1$$ where $N$ can be expressed as a sum of three fourth powers: $$N = a^4 + b^4 + c^4$$ for integers $a, b, c$.

Background: I see some from $\mathrm{A356496}$ for squarefree integers $k$ such that $x^4 - ky^2 = 1$ has a nontrivial solution. It is a little different and I couldn't find any solutions from it. Also for ratio here.

Question: Does there exist any integer $N = a^4 + b^4 + c^4$ such that $x^4 - Ny^4 = 1$ allows a non-trivial integer solution $(x, y)$ with $y \neq 0$?

I have performed computer searches for $0 \leq a, b, c \leq 20$, but found no examples by PARI.