  We provide sharp sufficient criteria for an integral $2$-varifold to be
induced by a $W^{2,2}$-conformal immersion of a smooth surface. Our approach is
based on a fine analysis of the Hausdorff density for $2$-varifolds with
critical integrability of the mean curvature and a recent local regularity
result by Bi-Zhou. In codimension one, there are only three possible density
values below $2$, each of which can be attained with equality in the Li--Yau
inequality for the Willmore functional by the unit sphere, the double bubble,
and the triple bubble. We show that below an optimal threshold for the Willmore
energy, a varifold induced by a current without boundary is in fact a curvature
varifold with a uniform bound on its second fundamental form. Consequently, the
minimization of the Willmore functional in the class of curvature varifolds
with prescribed even Euler characteristic provides smooth solutions for the
Willmore problem. In particular, the "ambient" varifold approach and the
"parametric" approach are equivalent for minimizing the Willmore energy.
