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You realize that same kind of 'we don't know if it'll work since it hasn't been shown to' argument has been made in regards to another character on this wiki, right? This case right now is the same as his was at the time; we go by what we've been shown.
And right now, nothing we've been shown suggests that a White Walker can't be killed (or even simply worn down) by A) a normal Wall level blade stabbing them in the face/chest/etc, B) a blast of fire equivalent to almost 3 grams of TNT in power (walking through typical flames is nothing compared to that), or C) a telekinetic blast equivalent to over 1 kilogram of TNT in power. The absence of evidence is not the evidence of absence, and should never be taken as such.
And right now, nothing we've been shown suggests that a White Walker can't be killed (or even simply worn down) by A) a normal Wall level blade stabbing them in the face/chest/etc, B) a blast of fire equivalent to almost 3 grams of TNT in power (walking through typical flames is nothing compared to that), or C) a telekinetic blast equivalent to over 1 kilogram of TNT in power. The absence of evidence is not the evidence of absence, and should never be taken as such.