Light-emitting diode luminous intensity at sub-mA currents
Not many manufactures will specify the luminous intensity at low levels because many LEDs have a minimum required current. I have tested some high power 3W LEDs at low currents and the LEDs begin to flicker and turn off at about 10 mA. 

There are some LEDs with high luminous intensity and low maximum current (e.g. 20-50 mA) that will emit a measurable intensity at very low currents. One such LED is the high efficacy OSRAM LA E63F which targets the automotive industry.  This amber LED has a luminous flux (QV) of 12.7 candela where IV:30°.

At very low current the illuminance (EV), measured in lux (lumens/m²), is 1% of the illuminance at 50 mA.   




  8) In the range where the line of the graph is broken, you must expect
  higher differences between single LEDs within one packing unit.
  
  7) Due to the special conditions of the  manufacturing processes of LED,
  the typical data or calculated correlations of technical parameters
  can only reflect statistical figures. These do not necessarily
  correspond to the actual parameters of each single product, which
  could differ from the typical data and calculated correlations or the
  typical  characteristic line. If requested, e.g. because of technical
  improvements, these typ. data will be changed without any further
  notice.





  At higher currents, the common assumption is a linear dependency. I
  doubt linearity is maintained at much lower currents,


It's actually the opposite. The current vs. QV is linear at low currents and drops off as the current rises. 



As you can see the Cree XP3G has a test current of 350 mA at 100% of rated luminous flux (QV). At 700 mA (2X) the QV is less than 200%.  Whereas at 50% test current (175 mA) the QV is 50%.  

You can assume the luminous flux will drop at a linear rate as current decreases from the published test current within the operating range of values.