Hadamard-type conjecture on graph and level-set concavity

Let $\Omega$ be a planar $C^2$ domain and let $v\in C^2(\overline{\Omega})$ be positive with zero boundary values. Write $\mathcal G=\operatorname{graph}(v)$, and define

$$G_v=v_{xx}v_{yy}-v_{xy}^2,$$ $$L_v=v_y^2v_{xx}-2v_xv_yv_{xy}+v_x^2v_{yy}.$$

Hadamard-type conjecture. (i) $\mathcal G$ is concave if and only if $G_v\geq 0$ for every $x\in\Omega$. (ii) $v$ is $-\infty$-concave, meaning that $\mathcal G$ has convex level sets $\{x\in\Omega:v(x)>c\}$, if and only if $L_v\leq 0$ for every $x\in\Omega$.

These equivalences relate concavity of the graph and convexity of its superlevel sets to differential inequalities involving the Hessian of $v$. The source presents them as conjectural because a proof was not supplied; no resolution is given in the provided text.