Density Bounds in Hua's Theorem for Prime Squares

We prove a density version of Hua’s theorem in the five-variable case: if $\mathcal{P}$ is a subset of the primes with relative lower density greater than $0.82$, then every sufficiently large $n\equiv 5(\mathrm{m}\mathrm{o}\mathrm{d}24)$ is a sum of five squares of primes from $\mathcal{P}$.