You're wrong. Conoco already has the first of the LNG tankers built and operating, but not in Alaska, yet. The Conoco LNG fleet were intended to move gas from Alaska to the west coast, but BP and Exxon seemingly talked them out of it for a while. The article on this subject in the WSJ includes the information that Conoco had reached an agreement with the state officials for the Valdez LNG pipeline.
Neither ConocoPhillips, nor anyone else in the World, has a LNG Tanker that can move LNG from Alaska to the US West Coast due to the Jones Act. No US ship builder makes a LNG Tanker so no company could own one. ConocoPhillips has multiple LNG interest around the world and any Tanker Purchase would not have been intended for Alaska to US West Coast.