The title says it all. Here comes the relevant text:
Challenges: Choose 1 of your participating STARK characters to get +2 STR until the end of the challenge.
Endless Endurance: boost on a character's STR or on the total STR in a challenge?
The character you choose "gets" the +2 STR.
That's what I was thinking but others made me doubt (a little). Thanks for the answer.