I had a walk on this path today.
I can confirm that path & signs follows the north side of the ditch and doesn't go across the stream.
I'll contact Lincolnshire CC PRoW team about the status, route & amendments to this PRoW since the signed route is  different to that on their map & OS.
Trace: https://www.openstreetmap.org/user/Thomas%20Jarvis/traces/3552100
Photos: http://jervi.uk/stuff/AshbyFootpath.pdf
I'll update this note once I get a response.