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If we assume the conjecture that 6 is the only even number such that F_2n is the sum of two squares then 2 cannot divide n. Then we also have F_2n=F_n*L_n and so if F_2n is the sum of two squares F_n is the sum of two squares since n is odd so F_2n is the sum of two squares if L_n is the sum of two squares. Also if N is odd and F_2N is the sum of two squares then L_n is the sum of two squares. So if we assume the question is equivalent to the following are there an infinite number of odd n such that L_n is the sum of two squares.

If we assume the conjecture that $6$ is the only even number such that $F_{2n}$ is the sum of two squares then $2$ cannot divide $n$. Then we also have $F_{2n}=F_n*L_n$ and so if $F_{2n}$ is the sum of two squares $F_n$ is the sum of two squares since n is odd so $F_{2n}$ is the sum of two squares if $L_n$ is the sum of two squares. Also if $n$ is odd and $F_{2n}$ is the sum of two squares then $L_n$ is the sum of two squares. 
So if we assume the question is equivalent to the following are there an infinite number of odd $n$ such that $L_n$ is the sum of two squares.

We have if F_2n is the sum of two squares 2 cannot divide n. We also have F_2n=F_n*L_n and so if F_2n is the sum of two squares F_n is the sum of two squares since n is odd so F_2n is the sum of two squares if L_n is the sum of two squares. Also if N is odd and F_2N is the sum of two squares then L_n is the sum of two squares. So the question is equivalent to the following are there an infinite number of odd n such that L_n is the sum of two squares.

If we assume the conjecture that 6 is the only even number such that F_2n is the sum of two squares then 2 cannot divide n. Then we also have F_2n=F_n*L_n and so if F_2n is the sum of two squares F_n is the sum of two squares since n is odd so F_2n is the sum of two squares if L_n is the sum of two squares. Also if N is odd and F_2N is the sum of two squares then L_n is the sum of two squares. So if we assume the question is equivalent to the following are there an infinite number of odd n such that L_n is the sum of two squares.

We have if F_2n is the sum of two squares 2 cannot divide n. We also have F_2n=F_n*L_n and so if F_2n is the sum of two squares F_n is the sum of two squares so F_2n is the sum of two squares so the question is equivalent to the following are there an infinite number of odd n such that L_n is the sum of two squares.

We have if F_2n is the sum of two squares 2 cannot divide n. We also have F_2n=F_n*L_n and so if F_2n is the sum of two squares F_n is the sum of two squares since n is odd so F_2n is the sum of two squares if L_n is the sum of two squares. Also if N is odd and F_2N is the sum of two squares then L_n is the sum of two squares. So the question is equivalent to the following are there an infinite number of odd n such that L_n is the sum of two squares.