After took the Always Winter quest from Jeor Mormont, when I found the White Walkers' party, they were fighting with Mance Rayder's party, and I saved game here.
(https://img.atwiki.jp/awoiafwb/attach/1/228/ww1a.jpg)
After I joined the battle to help Rayder, regardless of the result (win/lose/retreat), state of the quest doesn't progress, and the text "... successfully completed ... Talk to ..." is not added in the quest window though the main text says "Return to ... once you have interacted with and assessed ...".
It should be difficult for players to know whether the quest has been completed or not. Some player may go back to Jeor.
(Of course, if I challenge the fight after Mance Rayder's party has been defeated and assimilated into WW, the quest ends successfully at a contact.)
Related lines:
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qstr_You_must_gather_a_sm
You must gather a small force and possibly a Valyrian weapon and scout deep within the lands of always winter. Return to the Lord Commander once you have interacted with and assessed the threat.
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qstr_This_quest_has_been_s
This quest has been successfully completed. Talk to {s59} to claim your reward.
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mno_pre_join_help_defenders
Rush to the aid of the {s1}. ({s21})
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- Game version (and steam or downloaded):
AWoIaF v9.4-developer_version 5, Warband v1.174 downloaded
(Perhaps since v9.3 or earlier version)