Answer

If the one who caused the damage says that the damage was not there beforehand and that he caused it, then he is certainly liable. That is an admission of a litigant. But plainly, the case here is one where the damager also does not know. He only knows that he caused damage. If they simply do not know, it seems obvious that he must pay. But if the injured party is genuinely uncertain (and not merely lacking information), then it indeed seems that this is a case of an uncertain claim, and one cannot extract payment on the basis of uncertainty. An uncertain claim cannot extract from one in possession, even if the one in possession also claims uncertainty.
Perhaps according to the understanding of some later authorities (Acharonim), that the burden of proof is on the damager even though he is the one in possession, because there was negligence in guarding and now the burden of proof is on him in order to be exempt, one could say that the damager is liable. But this can be rejected on two counts: 1. Perhaps this was said only where damage definitely occurred, but there is some factor for exemption — not where there is doubt whether any damage was caused at all. In that case he would be liable only according to Abaye in the Rif, Bava Metzia 36, that if it began with negligence and ended with an unavoidable accident, he is liable even if the accident is unrelated to the negligence. That implies liability for the negligence itself, regardless of whether it caused the damage. 2. That position is itself not a reasonable one.
Therefore, in my opinion, he is exempt if he himself does not know whether he caused damage.