A chain of ${\mathbb{C}}^{\ast }$-flips of the moduli spaces of $\mathcal{O}$-twisted rank 2 constrained framed Hitchin pairs on a smooth curve

Let $X$ be a smooth complex projective curve. We prove that there exists a surjective commutative forgetful diagram from the chain of ${\mathbb{C}}^{\ast }$-flips of the moduli spaces of ${\mathcal{O}}_X$-twisted rank 2 constrained framed Hitchin pairs on $X$ to the chain of ${\mathbb{C}}^{\ast }$-flips of the moduli spaces of rank 2 framed modules on $X$.