Isotropic realizability conjecture for periodic gradients

Let $Y$ be the unit cube of $\mathbb R^d$. Let $u\in C^3(\mathbb R^d)$, with $d\geq 2$, be such that $\nabla u$ is $Y$-periodic. Assume that the critical points of $u$ are isolated, and that conditions \refe{C1Rd}. and \refe{C2Rd}. hold.

Isotropic realizability conjecture. Then $\nabla u$ is isotropically realizable in the torus with a positive conductivity $\sigma$ such that $\sigma,\sigma^{-1}\in L^\infty_\sharp(Y)$.

This conjecture concerns the existence of a bounded, uniformly positive scalar conductivity making a periodic gradient field realizable in the torus. The supplied text gives no resolution status, so it remains open.