Equality of the ergodic control eigenvalues

Let $u^*$ be the convex, $C^{1,1}(\mathbb{R}^n)$ solution associated to the eigenvalue $\lambda^*$, and define

$$\lambda_+:=\inf_{\psi}\sup_{|D\psi(x)|<1}\{\Delta\psi(x)+f(x)\},$$

where the infimum is over the relevant $C^2(\mathbb{R}^n)$ supersolutions with $\liminf_{|x|\to\infty}\psi(x)/|x|\geq 1$. Equality of the ergodic control eigenvalues. One should have

$$\lambda^*=\lambda_+.$$

The paper proves $\lambda^*\leq\lambda_+$ and proves the reverse inequality if a suitable $C^2(\mathbb{R}^n)$ supersolution with eigenvalue $\lambda^*$ exists. It conjectures that this additional existence assumption is unnecessary, motivated by the expected $C^2$ regularity of $u^*$ on the set where $|Du^*|<1$.