Bleg re Rules re Rights Offerings and Shares Held Short

Having a hard time getting a straight answer on this one. Let’s say you hold 1,000 shares of a stock short. The company announces that it’s doing a rights offering, and for every share a holder owns, you can buy a newly issued share directly from the company for 90% of the closing price on December 31st. You hold the shares short through the ex-date of the rights offering. The underlying owner of the shares you’ve borrowed decides to exercise his rights, and requests 1,000 new shares at 90% of the December 31 price. Who is responsible for providing these new shares to the owner, the company that issued the rights, or you? Please only respond if you have real expertise on this.

UPDATE: A little more research seems to suggest that the short borrower is obligated to deliver the rights to the owner, but then the owner gets to exercise, or sell, the rights as he see fit. So the short borrower is on the hook, at most, for the value of the rights. Correct?