It is my understanding that the torus that minimizes the Willmore energy is not yet known (this from a sentence in a 2005 paper by John Sullivan). Willmore conjectured that the Willmore energy for any smooth, immersed torus is $\ge 2 \pi^2$. From what I can gather, this energy is achieved by a standard rotationally round torus, derived from the Clifford torus.

Assuming the problem remains open, my question is:

Are there any viable candidates for different torus Willmore minimizers, or does the evidence point to the one just mentioned?

I'm wondering if the problem remains open because all strange, unlikely alternatives have not been ruled out, or because there are actually some viable alternatives. Likely the right reference I am not finding would suffice. Thanks!

**Addendum**(

*2Sep13*). As Renato Bettiol first pointed out below, the Willmore conjecture has been solved by Fernando Marques and André Neves. They posted a 96-page paper to the arXiv:

Fernando Marques and André Neves. "Min-Max theory and the Willmore conjecture." arXiv:1202.6036 (2012). Updated March 2013.

_{The Willmore Torus. Image by Tom Banchoff, cited in Morgan article.}

The result was hailed in a Huffington Post article by Frank Morgan:

Frank Morgan. "Math Finds the Best Doughnut." Huffington Post, 2 April 2012.

And there is a very nice description by Dana Mackenzie in an article that also describes the related resolution of the Lawson conjecture by Simon Brendle:

Dana Mackenzie. "What's Happening in the Mathematical Sciences." Vol. 9. American Mathematical Soc., 2013. (AMS link)

(*7Sep13*). One more remark. It remains open what might be the 2-holed or
3-holed Willmore surface of minimal bending energy. Dana says (p.28),

..., it is difficult to know what even to conjecture about such surfaces.

double-bubble conjecture, where it seemed clear what the solution must be, but it was difficult to rule out strange partitions of space. – Joseph O'Rourke Mar 20 '11 at 0:26