Let $\mu,\nu$ be two distributions on the same discrete space. Is it true that

$$\mathrm{H}\left(\frac{\mu+\nu}{2}\right) \ge \mathbb{E}_{xy}-\log\left(\frac{\sqrt{\mu(x)\nu(y)}}{2} + \frac{\langle\mu, \nu\rangle}{2}\right)$$

where $x\sim\mu$ and $y\sim\nu$ independently. Here $\mathrm{H}(\rho) = -\sum_{z}\rho(z)\log\rho(z)$ is the Shannon's entropy function and $\langle\mu,\nu\rangle =\sum_z{\mu(z)\nu(z)}$.

Note that if $\langle\mu,\nu\rangle = 0$, then both LHS and RHS are $\frac{\mathrm{H}(\mu)+\mathrm{H}(\nu)}{2}+1$, so the inequality holds true.

On the other extreme, if $\mu = \nu$, then by convexity of $z\mapsto-\log z$, we have $$\frac{1}{2}\mathrm{H}(\mu) + \frac{1}{2}\mathrm{H}_2(\mu) \ge \mathrm{RHS}$$ where $H_2$ is the second order Renyi entropy which is never bigger than the Shannon entropy. Since LHS is $\mathrm{H}(\mu)$ in this case, this shows the inequality is again true.

I can also verify the inequality when $\mu,\nu$ are flat distributions.

I wonder if this holds true for all $\mu, \nu$.