We study the relative fluctuations of the link overlap and the square
standard overlap in the three dimensional Gaussian Edwards-Anderson model with
zero external field. We first analyze the correlation coefficient and find that
the two quantities are uncorrelated above the critical temperature. Below the
critical temperature we find that the link overlap has vanishing fluctuations
for fixed values of the square standard overlap and large volumes. Our data
show that the conditional variance scales to zero in the thermodynamic limit.
This implies that, if one of the two random variables tends to a trivial one
(i.e. delta-like distributed), then also the other does and, by consequence,
the TNT picture should be dismissed. We identify the functional relation among
the two variables using the method of the least squares which turns out to be a
monotonically increasing function. Our results show that the two overlaps are
completely equivalent in the description of the low temperature phase of the
Edwards-Anderson model.