Define LOGLOG as the class of languages which can be computed in space O(loglog n) by a deterministic Turing machine (with two-way access to the input). Similarly define NLOGLOG as the class of languages which can be computed in space O(log log n) by a non-deterministic Turing machine (with two-way access to the input). Is it really not known that these classes differ?
I could only find some older surveys and a theorem that if they equal then L=NL (which is not just a trivial padding argument!), but somehow I feel that separating these classes cannot be that hard. Of course I might be completely wrong, but if every second bit of the input is the numbers from 1 to n in increasing order in binary, separated by some symbols, then the machines can already learn loglog n and with every other second bit we can input a problem that can fool a deterministic machine but not a non-deterministic one. I don't see yet exactly how this could be done but feels like a possible approach, as with this trick we can basically input a depth log n binary tree along with its structure instead of the usual linear tape.