Open problem in mathematics

$\large \int_0^1 \int_0^1 \cdots \int_0^1 \left\{ 1\div \sum_{n=1}^k x_n \right\} ^m \, dx_1 \; dx_2 \cdots dx_k$ Calculate in closed form, the integral above for $k\geq 3$ and $m\geq2$ .

Notation: $\{ \cdot \}$ denotes the fractional part function.

Clarification: There are $k$ integrals.

This is a part of the set Formidable Series and Integrals

#Calculus

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$