Quasiconvexity conjecture for radially symmetric effective Hamiltonians

Assume that $H(p)=\varphi(|p|)$, where $\varphi:[0,\infty)\to\mathbb{R}$ satisfies (H7), and let

$$0=s_0<s_1<\cdots<s_{2m}<\infty=s_{2m+1}$$

be the associated sequence, with $M_i=\varphi(s_{2i-1})$ and $m_j=\varphi(s_{2j})$. Suppose also that $\varphi(0)=\min\varphi=0$ and that $V\in C(\mathbb{T}^n)$. Let $\overline{H}$ denote the effective Hamiltonian associated with $H(p)-V(x)$. Quasiconvexity conjecture. If

$$\operatorname{osc}_{\mathbb{T}^n}V=\max_{\mathbb{T}^n}V-\min_{\mathbb{T}^n}V\geq\max_{i,j}(M_i-m_j),$$

then $\overline{H}$ is quasiconvex. When $n=1$, this conjecture is known, while the multidimensional case is described as challenging and remains open.