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 $n$?

For $n\geq 3$ the inequality can go either way.
Take $x_1=L\to \infty$ while others are equal to $1$ to get
$$LHS>\frac L {2\sqrt n}+\frac 1 2 L^{1/n}>\frac L n+O(1)=RHS$$
which holds for $n\geq 4$. For $n=3$, consider a triplet $(8,1,1)$ to get
$$LHS=\frac 1 2\Big(\sqrt{\frac {64+2} 3}+2\Big)>\frac {10} 3=RHS$$
To get $LHS<RHS$ for any $n\geq 2$, let $x_1\to 0$ while others are equal to $1$:
$$LHS\to\frac 1 2\sqrt\frac{n-1}n<\frac {n-1}n\leftarrow RHS$$
