The question here is, "necessary for what?" and the answer that applies equally to Henry and Charles Gould is, "fulfilling the ambition." I believe Conrad is saying here, "the demand of a dream-ideal to fulfill itself is a stronger aspect than the dream-ideal's object, which can change from person to person and age to age."

Note the phrase "conviction of" -- this too is cast in a subjective light, denying the existence of any objective necessity.