Sun's prime–Pell-number representation conjecture

Let $P_0=0$, $P_1=1$, and $P_{n+1}=2P_n+P_{n-1}$ for $n=1,2,3,\ldots$ be the Pell sequence. Sun's prime–Pell-number conjecture. Every integer $n>5$ can be written as the sum of an odd prime, a Pell number, and twice a Pell number; equivalently,

$$n=p+P_s+2P_t.$$

The two Pell numbers can further be required to be positive. The source describes this as an analogue of the prime–Fibonacci–Fibonacci conjecture and reports computational verification up to $5\times10^{13}$, but gives no proof or disproof.