  Let $d\ge 2$ and $n\ge d$ with $(d,n)\notin \{(2,2),(3,3)\}$. We consider
homogeneous Diophantine equations of degree $d$ in $n+1$ variables and whether
they have solutions in the primes. In particular, we show that a certain
local-global principle holds for almost all such equations, following on from
previous work of the author arXiv:2305.06306. We do this by adapting the
methods of Browning, Le Boudec and Sawin (Annals, 2023).
