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Simple solution. Wait to be super charged or for heavy ammo and then equip TLW. When you die after using your super or all of your heavy ammo, the unequip TLW. The deaths only set you back if TLW is currently equipped, meaning you're actively carrying it. You don't have to get strictly kills with TLW for progress to count.
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Thanks man, will try again tomorrow
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Use the sunbreaker. Way better chance of getting progress. I kinda struggled with that quest for a day or so, but I got my Chaperone regardless.