How large is the sum of the quadratic and the geometric mean? I conjecture the following inequality:

$$\frac{\sqrt{\frac{\sum_{i=1}^n x_i^2}{n}}+\sqrt[n]{\prod_{i=1}^n  x_i}}{2} \leq \frac{\sum_{i=1}^n  x_i}{n}$$ for all real $x_1, x_2, \cdots, x_n > 0$

I am able to prove it when $n=2$. 
Apply the power mean inequality between the $\frac12$power-mean and the arithmetic mean to $\frac{k^2+1}{2}$ and $k$ to obtain the following. 
Multiply by $x_1$ and let $x_2=kx_1$ to obtain the result.
$$\left(\frac{\sqrt{\frac{k^2+1}{2}}+\sqrt{k}}{2}\right)^2 \leq \frac{\frac{k^2+1}{2}+k}{2}$$
$$\left(\sqrt{\frac{k^2+1}{2}}+\sqrt{k}\right)^2 \leq k^2+1+2k$$
$$\sqrt{\frac{k^2+1}{2}}+\sqrt{k} \leq k+1$$
$$\sqrt{\frac{k^2x_1^2+x_1^2}{2}}+\sqrt{kx_1^2} \leq kx_1+x_1$$
$$\sqrt{\frac{x_2^2+x_1^2}{2}}+\sqrt{x_1x_2} \leq x_1+x_2$$
$$\frac{\sqrt{\frac{x_2^2+x_1^2}{2}}+\sqrt{x_1x_2}}{2} \leq \frac{x_1+x_2}{2}$$
I haven't made much progress with other $n$. I tried using the power mean inequality $\sqrt{\frac{\sum_{i=1}^n x_i^2}{n}} \leq \sqrt[n]{\frac{\sum_{i=1}^n x_i^n}{n}}$ for $n \geq 2$ to get the type of the radical the same on both sides, but that wouldn't work. 
The conjecture is my own. Can someone prove or disprove it for general $n$?

Your inequality is false for $n\ge 3$ : A counterexample is 
$$(x_1,x_2,\cdots,x_{n-1},x_n)=(1,1,\cdots,1,2)$$
For $(x_1,x_2,\cdots,x_{n-1},x_n)=(1,1,\cdots,1,2)$, we have
$$\frac{\sqrt{\frac{n-1+2^2}{n}}+2^{1/n}}{2}\le \frac{n-1+2}{n}.$$
This is equivalent to 
$$2+2x-\sqrt{1+3x}-2^x\ge 0$$
where $0\lt x=\frac 1n\le 1$.
Let $f(x)=2+2x-\sqrt{1+3x}-2^x.$ Then, 
$$f'(x)=2-\frac{3}{2\sqrt{1+3x}}-2^x\ln 2.$$
For $0\lt x\le \frac{1}{10}$,
$$f'(x)=2-\frac{3}{2\sqrt{1+3x}}-2^x\ln 2\lt 2-\frac{3}{2\sqrt{\frac{3}{10}+1}}-\ln 2\lt 0.$$
With $f(0)=0$, we know that $f(x)\lt 0$ at least for $0\lt x\le \frac{1}{10}$.
Hence, we know that your inequality is false for $n\ge 10$. Checking each of $n=3,4,\cdots,9$ gives that your inequality is false for $n\ge 3$.
