specify maxspeed for each lane

Just add the tags as suggested in our docu wiki at Key:maxspeed#Lanes – with any editor. In addition, the standard street form template in JOSM has pre-configured fields for maxspeed:forward and maxspeed:backward (but not for the single lanes).

Bonus: For QA and ease of tagging there are two maxspeed addon map styles available for JOSM, see https://josm.openstreetmap.de/wiki/Styles#Availablestyleslist and search for maxspeed. You can install them directly from JOSM's options dialogue.

The above answer is the simplest. If you will frequently use that tag in JOSM, there are lane templates that I believe have places to enter the speed limit in each lane. Mapbox guide to mapping turn lanes

In JOSM, go to Edit > Preferences.
Then in preferences click on the Map Settings tab (The grid with a circle).
In the Map Paint Style button, check the Lane and Roads attributes.
Then click the Tagging Preset button and check the Lane attributes.

While editing a street just highlight it and go to Presets > Lanes and I think one of those has the maxspeed:lanes preset. Happy Tagging!