Mean and minimum of independent random variables

We show that any pair X, Y of independent non-compactly supported random variables on [0, ∞) satisfies liminf_m →∞ℙ(min (X,Y) > m | X + Y > 2m.) = 0. . We conjecture multi-variate and weighted generalizations of this result, and prove them under the additional assumption that the random variables are identically distributed.