The principle of conservation of information states that search algorithms cannot consistently outperform random sampling unless the search space has a helpful structure that can be exploited or the search algorithm benefits from information about the desired target. This poses a conundrum for unguided evolutionary searches: if these found the extraordinarily rare configurations of matter we see in life, they must have received help of one of these kinds, but in that case, how can we view them as unguided searches? One possibility to be considered is that the search spaces most relevant to biological evolution just naturally possess features that are highly non-random in a helpful way. Recent work has argued that the mapping of geneotypes to phenotypes is such a helpful feature. Following the methods of others, we here model evolutionary paths in RNA sequence space to see how likely these are to find sequences that have pre-specified base-paired structures (target structures). In agreement with others, we find that short target structures (20 or 30 bases) chosen from a database of biological non-coding RNAs (fRNAdb) often are readily found by searches in RNA sequence space. This is explained by the fact that the database is strongly skewed toward structures commonly attained by random sequences. Nevertheless, by calculating active-information from thousands of evolutionary simulations, we show that unguided RNA search algorithms consistently underperform random sampling, as is expected from the conservation principle. Equally important, we argue that the connection between RNA sequences and their structures is so unlike the connection between the genotypes and phenotypes of living organisms that studies of the former do very little to advance our understanding of the latter. The unanswered challenge that the conservation of information poses for unguided evolution remains unanswered.