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I figure the answer is hidden in the definition of "marginal contribution to risk". I try to use the notation of the paper you linked.

Marginal contribution to risk is defined as:

$M_i = 2C_{ip}$ with $C_{ip}$ being the covariance between asset $i$ and the portfolio $p$. One can now argue that, since this covariance also depends on the other assets $j\neq i$, the value $M_i$ is implicitly connected to the other assets via the correlation structure.

On the other hand if all assets are uncorrelated, we have that $C_{ip}=C_{ii}w_i$, since $C_{ij}=0$ for $j\neq i$. Thus, the marginal contributions do not depend on the other assets. Therefore one can speak of a more "pure" contribution to variance in this case.

Finally, if the asset returns are independent, they are also uncorrelated.