Sun's Lucas and mixed Fibonacci–Lucas representation conjectures

Let $F_n$ and $L_n$ denote the Fibonacci and Lucas sequences, respectively. A sequence term is positive when its value is positive, and a term is odd when it is not divisible by $2$. Sun's Lucas and mixed Fibonacci–Lucas conjectures. (i) Every integer $n>4$ can be written as the sum of an odd prime, an odd Lucas number, and a positive Lucas number. For $k=2,3$, every integer $n>4$ can be written as $p+L_s+L_t^k$, where $p$ is an odd prime, $s,t\geqslant0$, and $L_s$ or $L_t$ is odd. (ii) Every integer $n>4$ can be written as the sum of an odd prime, a positive Fibonacci number, and twice a positive Fibonacci number, or half of a positive Fibonacci number. Also, every integer $n>4$ can be written as the sum of an odd prime, twice a positive Fibonacci number, and the square of a positive Fibonacci number. (iii) Every integer $n>4$ can be written as $p+F_s+L_t$, where $p$ is an odd prime, $s>0$, and $F_s$ or $L_t$ is odd. The source reports extensive computational checks, but these assertions remain open.