One aspect is the balancing (weighting) of the two losses: giving l2 a higher weight will prefer reducing it over achieving the same reduction of l1.
Another thing to check is whether l2 has actually achieved a minimum, egif the gradients of l2are (almost) zero. In that case you would want to revisit the modelling. As always, overfitting a single batch is a good sanity check here.
Best regards
Thomas