Let us have a tuple of strings $x_1,\ldots,x_n$, and some oracle $O$.
Assume that there is no information in $O$ about $x_i$'s, i.e.,
$I(x_1,\ldots,x_n;O)\approx0$.
Can any `informational' properties of $x_1,\ldots,x_n$  
essentially change under relativization conditional to the oracle $O$?
We discuss a formalization of this question and show that some special
cases of these general conjecture  hold. In particular, we prove that
if an oracle has very few mutual information with given strings,
then relativization conditional to this oracle cannot help to extract the
mutual information between these strings.