Soydan–Németh–Szalay conjecture on weighted power sums of Fibonacci numbers

Let $F_n$ denote the $n$th Fibonacci number, and let $p,q,k$ be positive integers. Consider the Diophantine equation

$$F_1^p+2F_2^p+\cdots+kF_k^p=F_n^q.$$

The trivial solutions are $F_1^p=1=F_1^q=F_2^q$ and $F_1^p+2F_2^p=3=F_4$.

Soydan–Németh–Szalay conjecture. Apart from these trivial solutions, the only solutions are

$$F_4^2=9=F_1+2F_2+3F_3,$$ $$F_8=21=F_1+2F_2+3F_3+4F_4,$$

and

$$F_4^3=27=F_1^3+2F_2^3+3F_3^3.$$

This conjecture is based on the cases solved in the paper and a computer search with $p,q,k\leq100$; the supplied text gives no resolution beyond that evidence.