Kusner's 16π-conjecture for sphere eversions

A sphere eversion is a regular homotopy $H:\mathbb{S}^2\times[0,1]\to\mathbb{R}^3$ connecting two round spheres of opposite orientation. Kusner's 16π-conjecture. There exists a sphere eversion $H$ satisfying, for every $t\in[0,1]$,

$$W(H_t)\leq 16\pi.$$

Every sphere eversion must reach Willmore energy at least $16\pi$ because it has a quadruple point at some time and the Li--Yau inequality applies. Whether the lower bound is attainable by an optimal sphere eversion is not resolved in the supplied text.