The inductance goes down with DC current. And it goes up with AC voltage (until it reaches a peak) due to permeability change of the iron lams (substantial 100 to 1 change, see below). The air gap moderates that 100 to 1 variation considerably, fortunately, but not entirely. The L meter only puts out a small AC voltage signal, so the iron permeability will be low.

Checking some Hammond 10H chokes with an L meter:

193M 300mA, 10H rated: 8.58H measured

193Q 500mA, 10H rated: 6.7H measured

193L 300mA, 5H rated: 4.19H

What is the DC current rating for the choke in question?

Results will depend on the AC signal level put out by the L meter. The L meter here puts out 1.24VAC at 93 Hz