What should the guardian of his deceased sister's children do regarding alleged debts of the deceased for which there is no proof, especially since the deceased denied having any debts, and there is a witness from the claimant's relatives?

Mere assertion is not enough to prove a right, for the Prophet, peace and blessings be upon him, said: "Were people to be given according to their claims, men would claim the lives and wealth of others. But the burden of proof is upon the claimant, and the oath is upon him who denies." A debt is established by the testimony of a just witness and the claimant's oath, or by the admission of the heirs. If the witness is not just, or the claimant refuses to take the oath, or the adult heirs do not admit it, then the right is not established by mere claim, and money should not be paid except with proof. Matters of disputes require direct verbal communication or referral to the judiciary.