Richard Gale and I have shown that once you grant:
- WPSR: for all contingent truths p, it is possible that p has an explanation,
- WWPSR: possibly WPSR is true.
Nec explains at least one contingent truth in every world. For Nec explains at least one contingent truth at w1 (as it explains them all there). So suppose for a reductio that it is a contingent proposition that Nec explains at least one contingent truth. Call that proposition e. Since it is contingent, and it is true at w1, Nec must explain e at w1. But arguably it would be circular for an exercise of explanatory efficacy to explain why there is at least one piece of explanatory efficacy.
Moreover, since Nec exists in every world, what could prevent his activity from having explanatory efficacy in some world w2? Whatever that is, it is something that he must have squelched in w1. So in w2, presumably he did not squelch that thing or event, and that seems to be a contingent truth at w2 that he had explanatory efficacy over.