Algebraicity conjecture for real symplectic del Pezzo surfaces of degree 1

Let a real symplectic del Pezzo surface of degree $1$ be equipped with a $J$-holomorphic structure satisfying the Bertini condition. Algebraicity conjecture. Each deformation class of such surfaces contains a real algebraic del Pezzo surface of degree $1$. The conjecture would show that every relevant deformation class in the symplectic setting has an algebraic representative; the source suggests that its proof may be achievable using methods from earlier work, but does not establish it.