Different from Farron's Undead Legion, the second Lord of Cinder Aldrich didn't intend to link the fire. Instead, it hoped that the first flame would die out.
It even foresaw the deep ocean era after the first flame extinguished. In this era, an ocean would swallow the land, and darkness would swallow light.
However, Pontiff Sulyvahn hoped that the Profaned Flame would burn up the whole world.
In short, the former thought that the end would come sooner or later and would rather protect himself, and the latter intended to take the throne with his blazing ambition and conspiracies and finish the last gods, ending the era of fire by allowing darkness to devour the light.
They were waiting for the First Flame to die out and the darkness to fall.
It was impossible to ask them to link the fire.
Besides, they even encouraged more unkindled to join the Church of the Deep and kill the supporters of Firelink.
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