Consider the semilinear elliptic equation Δu+u−u3=0 on Rn, where Δ denotes the Laplacian. A function u:Rn→R is called a bounded classical solution if u is twice continuously differentiable, there exists a constant C such that ∣u(x)∣≤C for all x∈Rn, and u satisfies Δu(x)+u(x)−u(x)3=0 at every point x∈Rn. A bounded classical solution u is said to have positive first partial derivative if ∂1u(x)>0 for all x∈Rn, meaning the directional derivative in the direction of the first standard basis vector is strictly positive everywhere. The solution u is said to have hyperplane level sets if for every value y in the range of u, there exists an affine subspace S⊆Rn of codimension 1 (that is, whose direction has dimension n−1) such that the level set u−1({y})={x∈Rn:u(x)=y} coincides with S. De Giorgi's conjecture asserts that in dimension n≤8, every bounded classical solution with positive first partial derivative has hyperplane level sets. For n=4, does this rigidity property hold? In other words, does every bounded classical solution u:R4→R to Δu+u−u3=0 satisfying ∂1u>0 everywhere necessarily have hyperplane level sets?