  It is known that the eigenfunctions of a random Schr\"odinger operator on a
strip decay exponentially, and that the rate of decay is not slower than
prescribed by the slowest Lyapunov exponent. A variery of heuristic arguments
suggest that no eigenfunction can decay faster than at this rate. We make a
step towards this conjecture (in the case when the distribution of the
potential is regular enough) by showing that, for each eigenfunction, the rate
of exponential decay along any subsequence is strictly slower than the fastest
Lyapunov exponent, and that there exists a subsequence along which it is equal
to the slowest Lyapunov exponent.
