Main conjecture on limited prime power sums

Let $n$ be a natural number greater than $23$. A prime power is a number $P^k$ where $P$ is prime and $k\geq 2$ is an integer. Main conjecture on limited prime power sums. Every natural number $n>23$ can be expressed as a sum

$$n=\sum_{i=1}^{m}P_i^{k_i},$$

where each $P_i$ is prime, each $k_i\geq 2$ is an integer, and $2\leq m\leq 5$. The conjecture is supported by exhaustive computation through $10^7$ and sampled checks at larger values, but no proof is given; establishing the bound of five summands remains open.