I have a problem that is reducible to (efficiently) determining the reachability of a node in a fully dynamic planar digraph.

I'm aware of "A fully dynamic data structure for reachability in planar digraphs" which provides $O(n^{2/3} \log n)$ query with a $O(n)$-space data structure.

Can this be / has this been bettered?

If all my queries have the same source node, is there a more efficient (in time/space/both) way?

Are there any other related literature that deals with more efficient queries albeit with more restrictions imposed on the digraph?

Thanks!