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Though it has been two years since you asked the question, you might be interested in our coming paper. Recently, we are working with an essentially equivalent problem and able to show that there is a $k$-wise linearly independent set of $\Omega(2^{n/[k/2]})$ vectors. We also prove some non-trivial lower bound for general $q$, that beat the gready trivial lower bound.
How about if we replace arithmetic mean with geometric mean, namely, do there exist $A, B > 0$ such that $ X = A \sharp B$ and $Y = \left(\dfrac{A^p+B^p}{2}\right)^{1/p}$ with $p > 1$?
