An integral inequality for Gaussian sequential decision making

Let $N=2$ and suppose the conditional densities of the observation $Y$ given hypotheses $H=0$ and $H=1$ are

$$f_{Y\mid H}(y\mid 0)=\frac{1}{\sqrt{2\pi\sigma^2}}\exp\left(-\frac{y^2}{2\sigma^2}\right),$$ $$f_{Y\mid H}(y\mid 1)=\frac{1}{\sqrt{2\pi\sigma^2}}\exp\left(-\frac{(y-h_1)^2}{2\sigma^2}\right),$$

where $\sigma$ and $h_1$ are arbitrary positive numbers, and let $\lambda$ be a real number. The Gaussian integral inequality. If $\lambda<\frac{h_1}{2}$, then

$$\int_{-\infty}^{\lambda}\exp\left(-\frac{y^2}{2}+\lambda h_1\right)\,dy\int_{\lambda}^{\infty}\exp\left(-\frac{y^2}{2}+\lambda h_1\right)\,dy < \int_{-\infty}^{\lambda}\exp\left(-\frac{y^2}{2}+y h_1\right)\,dy\int_{\lambda}^{\infty}\exp\left(-\frac{y^2}{2}+y h_1\right)\,dy.$$

This inequality is used in the analysis of the prior belief that minimizes the second agent's Bayes risk in a two-agent sequential decision problem with additive Gaussian observation noise. The surrounding discussion indicates that optimal prior beliefs can differ from the true prior probability, but the supplied text does not establish whether this asserted inequality is proved or remains open.