Fix $x\in \mathcal{X}$ and take the functions $$ \bar{t}(x)\equiv \sum_{y\in \mathcal{Y}}(\bar{g}(x,y)-\bar{h}(x,y))p(y) $$ $$ t(x)\equiv \sum_{y\in \mathcal{Y}}(g(x,y)-h(x,y))p(y) $$ where
$p(y)\in [0,1]$ for each $y\in \mathcal{Y}$ and $\sum_{y\in \mathcal{Y}}p(y)=1$
$\bar{g}, g, \bar{h}, h$ have the same domain $\mathcal{X}\times \mathcal{Y}$
$\bar{g}(x,y)\in [0,1]$, $\bar{h}(x,y)\in [0,1]$, $g(x,y)\in [0,1]$, $h(x,y)\in [0,1]$ for each $(x,y)\in \mathcal{X}\times \mathcal{Y}$
$\bar{g}(x,y)\geq g(x,y)$ and $\bar{h}(x,y)\geq h(x,y)$ for each $(x,y)\in \mathcal{X}\times \mathcal{Y}$
Consider the intervals $$ \bar{I}=[\min_{x\in \mathcal{X}} \bar{t}(x), \max_{x\in \mathcal{X}} \bar{t}(x)] $$ $$ I=[\min_{x\in \mathcal{X}} t(x), \max_{x\in \mathcal{X}} t(x)] $$
Should we expect that $I\subseteq \bar{I}$?