Is every odd integer $n>351$ expressible as $a^3+b^2+c$ with primes $a,b,c$?

Question

This conjecture

The set of numbers of the form $q_1+q_2^2+q_3^3+q_4^4+q_5^5$ where all $q_k$ are primes.

would be true if for every odd $n>351$ , there are primes $a,b,c$ with $a^3+b^2+c=n$.

Is this conjecture true , or is there at least a heuristic for it ?