Rational integrability criterion for diagonal hypersurface reciprocals

Let $n\in\mathbb{N}$ be fixed, let $m\geq 4$, and consider the rational function

$$f=\frac{1}{x_1^n+\cdots+x_m^n}.$$

Rational integrability conjecture. The function $f$ is rationally integrable with respect to $x_1,\ldots,x_m$ if and only if $n\neq m$. This extends the discussion of rational integrability and its interpretation in terms of exact differential forms. The supplied text does not establish the assertion or provide evidence resolving it, so its status remains open.