Towards a counter-example to a conjecture of De Giorgi in high dimensions

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Entire solutions of the Allen-Cahn equation and complete embedded minimal surfaces. (2009, background): “A counterexample was recently built for N ≥ 9 by us in [11, 12]; see also [6, 27].”
Saddle-shaped solutions of bistable diffusion equations in all of ℝ2m (2009, background): “1 (Jerison–Monneau [19]).”
Regularity of radial minimizers and extremal solutions of semilinear elliptic equations (2005, background): “and has been revisited recently in connection with a conjecture of De Giorgi (see [1,7,13] and references therein).”