By $C(|)$ denote conditional complexity.
By $CT(|)$ denote total conditional complexity.
For every n there exist two strings $x$ and $y$ of length $n$ such that $C(x|y) = O(1)$ but $CT(x|y) \ge n $.
It is proved there: http://arxiv.org/pdf/1204.0198.pdf pages 5-6.
It is easy to see that $x$ have the same information as $y$ (Let us denote it $ x \sim y$)
What is the information?
By $T_n$ denote number of total programs with length $< n$.( $T_n \sim N_k$ - number of integers with complexity $\le n$. $N_k \sim \Omega_n$ - the first $n$ bits Chaitin's number.)
Let $x=x_n$ and $y=y_n$ where $x_n$ and $y_n$ are the same as in article. Then $x \sim T_n$: we can play this computable game until Alice add $x_n$ - so we find $T_n$.
Question: let $x$ and $y$ have lengths $n$ , $C(x|y) = O(1)$ and $CT(x|y) \ge n $.
Is it true that $x \sim T_n$ necessarily?