Solvability of the equation $2a_{1}^{2} = a_{2}^{n} + a_{3}^{n} + a_{4}^{n}$ when $n \geq 5$ is prime? As a natural extension of the question titled
Solvability of $a_{1}^{2} = a_{2}^{n} + a_{3}^{n} + a_{4}^{n}$ when $n \geq 5$ is prime?,
I wonder if the equation
$$2a_{1}^{2} = a_{2}^{n} + a_{3}^{n} + a_{4}^{n}$$
is solvable for co-prime $a_i$ when $n \geq 5$ is prime?
The Darmon-Merel theorem asserts that $a_{1}^{2} \neq a_{2}^{n} + a_{3}^{n}$ for all $n \geq 3$ and all integers $a_{1}, a_{2}, a_{3}$ not all 
$0$, but I am not sure about the result when $2a_{1}^{2}$ is considered instead.
If there is already any known result in existence, bringing it in is more than welcome.

I do not know why co-prime solutions are more interesting and must these be mutual co-prime or co-prime altogether, but there are few mutual co-prime  ones $a_1,a_2,a_3<1000$ (not sure all of these found due to precision trunks):
$$\begin{align}4^5+239^5+659^5=2\cdot 7907819^2\\
40^5+ 617^5+ 633^5= 2\cdot 9773625^2\\
73^5+ 121^5+ 144^5= 2\cdot 212047^2\\
96^5+ 709^5+ 877^5= 2\cdot 18681029^2\\
161^5+ 380^5+ 479^5= 2\cdot 4077240^2\\
211^5+335^5+704^5=2\cdot 9422435^2\\
224^5+387^5+541^5=2\cdot 5271996^2\\
360^5+493^5+517^5=2\cdot 6004405^2\\
369^5+640^5+881^5=2\cdot 17957625^2\\
\end{align}$$
