Rev. Canon Paul Wancura, 87, was attacked and left to die inside his Oak Tree Lane house on March 19, 2018.
Police said the suspect also made off with jewelry, including a Lucien Piccard Seashark watch.
After having his arm amputated, Rev. Wancura succumbed to his injuries on April 16, 2018.
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The 87-year-old priest served the diocese for more than half a century in Setauket, Greenport and Shelter Island.
Suffolk County Crime Stoppers are offering a $10,000 cash reward for information leading to an arrest in the case. Anyone with information can submit an anonymous tip by calling 1-800-220-TIPS, texting “SCPD” and your message to “CRIMES” (274637) or by emailing www.tipsubmit.com. All messages will be kept confidential.