The $16\pi$ conjecture for sphere eversion

Let $\Omega$ be the set of paths in $\operatorname{Imm}(\mathbb{S}^2,\mathbb{R}^3)$ joining the standard $\mathbb{S}^2$ to the oppositely oriented sphere, and define

$$\beta_0=\inf_{\omega\in\Omega}\sup_{\vec{\Phi}\in\omega}W(\vec{\Phi}).$$

The $16\pi$ conjecture. The number $\beta_0$ is attained by a path $\omega\in C^0([0,1],\operatorname{Imm}(\mathbb{S}^2,\mathbb{R}^3))$ such that both restrictions $\omega|_{[0,1/2]}$ and $\omega|_{[1/2,1]}$ are Willmore flows, and $\omega(1/2)$ is the inversion of the Morin surface, a minimal surface with four planar ends. The passage describes this as a step toward the $16\pi$ conjecture and gives the lower bound $\beta_0\geq16\pi$, but does not establish the asserted attainment statement.