= Solution
Let $P$ be the state-by-asset <matrix> of discounted gains and identify $Y$ with its state <vector>. Fix $q^0\in\mathcal Q$. If $q\geq0$, $P^Tq=0$, and $q\ne0$, then
$$
q_\varepsilon=\frac{q+\varepsilon q^0}{\sum_i(q_i+\varepsilon q_i^0)}
$$
belongs to $\mathcal Q$. The assumption and passage to the <limit> imply $q^TY\geq0$ for every nonnegative $q\in\ker P^T$.
The <finite-state superhedging alternative> now gives $\theta$ with
$$
Y-P\theta\geq0.
$$
This residual cannot vanish identically, since then every $Q\in\mathcal Q$ would have $E_QY=q^TP\theta=0$. Hence it is strictly positive in at least one state, which has positive physical probability. This is the required inequality.
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