In general, the heavier the conversion coating, the poorer the conductivity. In the aerospace world: "MIL-C-5541, Class 3" is called out where electical conductivity is the primary consideration, and corrosion reistance is secondary.
As with MIL-C-5541, the relatively thin, Class 3 callout in B449 is identified as being used where electrical conductivity is the prime consideration. The ASTM specification identifies the Class 3 coating as "colorless", "decorative" and having "slight corrosion resistance" and "low electrical contact resistance".
. It has some good information regarding factors (in addition to coating thickness) that can affect conductivity. I'm not suggesting you specify an "aerospace" specification if you don't need to (use the ASTM Spec TVP cited if it meets your needs), but there's some good info in the MIL document.
A proprietary (sole source) non-chromated, "environmentally friendly" conversion coating that is reported to have good electrical conductivity properties is available from Sanchem (