Prime plus two positive cubes

Prove that every positive integer n greater than 308 can be represented as n=p+x^3+y^3, where p is an odd prime and x and y are positive integers.

Background

This is the cube analogue of the prime-plus-two-squares conjecture. The paper proves the assertion computationally for 308<n<1016, while the unrestricted statement remains conjectural.

References

If $n>308$ is a positive integer, then $n=p+x3+y3$, where $p$ is an odd prime and $x,y$ are positive integers.

— Computational results on sums of a prime with squares or cubes  (2609.20505 - Applegate et al., 17 Sep 2026) in Conjecture 1.4, Section 1