The leash does not render as shown below (just for the pig, not the other model which uses GeoEntityRenderer):

This image was taken using Fabric on 22w11a. To elucidate this issue, it seems like normal minecraft code renders the leash in (Mojang mappings) MobRenderer#renderLeash which is a private method. Conversely to geckolib as of now which does not have any rendering method/logic for leashes in GeoEntityRenderer. Is this feature planned to be added in the near, forseeable future?

This logic requires logic from Mob Entity and will require you to create a custom Renderer using that instead of living entity. This will not be be added to the base library