The $x-$coordinates of Pell equations and sums of two Fibonacci numbers II

Let $ \{F_{n}\}_{n\geq 0} $ be the sequence of Fibonacci numbers defined by $ F_0=0 $, $ F_1 =1$, and $ F_{n+2}= F_{n+1} +F_n$ for all $ n\geq 0 $. In this paper, for an integer $ d\ge 2 $ which is square-free, we show that there is at most one value of the positive integer $ x $ participating in the Pell equation $ x^2-dy^2=\pm 4 $ which is a sum of two Fibonacci numbers, with a few exceptions that we completely characterize.