Non-one-dimensional minimizer conjecture in dimension seven

Assume (A1): $\beta$ is non-negative and $C^1$, with $\textrm{supp}\,\beta=[0,1]$ and $\int_0^1\beta=1$. Let

$$J(u,E)=\int_E\left\{|\nabla u|^2+\Phi(u)\right\}\,dx,$$

where $\Phi'=\beta$. Dimension-seven non-one-dimensional minimizer conjecture. There exists a globally Lipschitz minimizer of the energy $J$ in $\mathbb{R}^7$ that is not one-dimensional.

This is motivated by the existence of non-flat minimal cones for the one-phase problem in dimension $7$. The claim is presented as a second natural open problem, and no resolution is given in the source.