Sun's prime–Fibonacci-power representation conjectures

Let $F_n$ denote the Fibonacci sequence. A Fibonacci number is positive when its value is positive, and an odd Fibonacci number is one not divisible by $2$. Sun's prime–Fibonacci-power conjectures. (i) Every integer $n>4$ can be written as the sum of an odd prime, a positive Fibonacci number, and the square of a positive Fibonacci number; one of the two Fibonacci numbers can further be required to be odd. (ii) Every integer $n>4$ can be written as the sum of an odd prime, a positive Fibonacci number, and the cube of a positive Fibonacci number; one of the two Fibonacci numbers can further be required to be odd. The source reports computational verification for these assertions in substantial ranges, but gives no proof or disproof; they remain open.