The Artin conjecture for three diagonal cubic forms

Let p be a prime, p ≠ 3or 7. Then any three diagonal cubic forms over the p-adics in at least 28 variables possess a common nontrivial p-adic zero. This verifies the Artin conjecture for three diagonal cubic forms over all p-adic fields with the possible exception of $\text{Q}$₃ and $\text{Q}$₁.