What are the obstructions to extending $L=SL$ to $L=NL$?
The central problem is that, on directed graphs, even a truly random walk doesn't hit all the vertices in expected polynomial time, let alone a pseudorandom walk.  The standard counterexample here is a directed graph with $n$ vertices ordered from left to right, where each vertex has an edge leading to the vertex to its right (except for the rightmost vertex, $t$), and each vertex also has an edge leading all the way back to the leftmost vertex, $s$.  To get from $s$ to $t$ by a random walk then takes ~$2^n$ time.  So, what is the small-space randomized algorithm for directed connectivity that we're hoping to derandomize, analogous to what Reingold did for $USTCON$?  (To put it another way, how do we show $RL=NL$, let alone $L=NL$?)  For directed connectivity, of course there's Savitch's algorithm, but that takes $O(\log^2 n)$ space, and for general graphs no one has managed to improve it for half a century, with or without the use of randomness.