Translation invariant quadratic forms in $8$ variables and dense sets of primes

Let $Q$ be a translation invariant indefinite quadratic form in $8$ variables. Let $\mathcal{P}$ be the set of all primes. Let $\mathcal{A}\subset \mathcal{P}\cap \{1,\dots ,X\}$. By developing the recent work of Green, we prove that subject to a mild condition on the form $Q$ there exist pairwise distinct primes $p_1,\dots ,p_8\in \mathcal{A}$ satisfying $Q(p_1,\dots ,p_8)=0$ provided that $$\frac{\mathrm{\mid }\mathcal{A}\mathrm{\mid }}{X\mathrm{/}\log X}{\gg }_Q(\log \log X{)}^{-c}$$ for some absolute constant $c>0$.