The second-derivative characterization of h-convexity

Let $I$ and $J$ be the domains appearing below, let $h:J\to(0,\infty)$ be a non-negative function such that

$$h(\alpha)\geq\alpha\quad\text{for all }\alpha\in(0,1),$$

and let $f:I\to\mathbb{R}$ be twice differentiable. A function $f$ is $h$-convex if it satisfies the corresponding $h$-convexity inequality.

Second-derivative characterization. The function $f$ is $h$-convex if and only if

$$f^{\prime\prime}(x)\geq 1-2h\left(\frac{1}{2}\right).$$

This conjecture proposes a necessary and sufficient second-derivative criterion for $h$-convexity, generalizing the familiar characterization of ordinary convexity by nonnegative second derivative. The supplied text does not establish the claim or indicate whether it has been resolved.