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For any integers $i$ and $j$ such as $1\le i<j\le6$, let $x_{ij}$ be a real number.

Is it true that, given the condition
$$\sum_{1\le i<j\le6}x_{ij}^2=1,$$ the sum $$\sum_{1\le i<j<k\le6}x_{ij}x_{ik}x_{jk}$$ is maximized when all the $|x_{ij}|$'s are the same?

For any integers $i$ and $j$ such as $1\le i<j\le6$, let $x_{ij}$ be a nonnegative real number.

Is it true that, given the condition
$$\sum_{1\le i<j\le6}x_{ij}^2=1,$$ the sum $$\sum_{1\le i<j<k\le6}x_{ij}x_{ik}x_{jk}$$ is maximized when all the $x_{ij}$'s are the same?