Functional $L_p$-Gardner–Zvavitch conjecture

Let $p\in[1,\infty)$, let $f,g:\mathbb{R}^n\to\mathbb{R}_+$ be centered integrable functions, let $\mu$ be a measure on $\mathbb{R}^n$, and let $t\in(0,1)$. Let $h:\mathbb{R}^n\to\mathbb{R}_+$ be measurable and suppose that, for every $x,y\in\mathbb{R}^n$ and every $0<\lambda<1$,

$$h\left((1-t)^{\frac{1}{p}}(1-\lambda)^{\frac{p-1}{p}}x+t^{\frac{1}{p}}\lambda^{\frac{p-1}{p}}y\right)\geq f(x)^{(1-t)^{\frac{1}{p}}(1-\lambda)^{\frac{p-1}{p}}}g(y)^{t^{\frac{1}{p}}\lambda^{\frac{p-1}{p}}}.$$

Functional $L_p$-Gardner–Zvavitch conjecture. There exists an absolute constant $C\geq1$ such that

$$\int_{\mathbb{R}^n}h(x)\,d\mu(x)\geq\frac{1}{C^p}\left[(1-t)\left(\int_{\mathbb{R}^n}f(x)\,d\mu(x)\right)^{p/n}+t\left(\int_{\mathbb{R}^n}g(x)\,d\mu(x)\right)^{p/n}\right]^{n/p}.$$

This is presented as a functional counterpart of the Gardner–Zvavitch conjecture. The paper says that it is proved in some special cases, while the existence of a universal constant in the stated generality remains open.