$d^+d^*\omega=0$ implies $dd^*\omega=0$ for a 2-form $\omega$ on a 4-manifold

Question

Let $\omega$ be a smooth 2-form on a closed oriented riemannian 4-manifold $X$. How can we show that $d^+d^*\omega=0$ (i.e. $dd^*\omega$ is anti-self-dual) implies $dd^*\omega=0$?

This statement is claimed in the proof of Claim 2.6 in https://web.ma.utexas.edu/users/pedrotti.riccardo/Riccardo_Pedrotti-Notes_on_the_paper_of_Bauer_about_Refined_SW_invariants.pdf. (In this note, it is written that a proof of this statement is given in one of the references but it is not available.)