= Solution
\b[Yes.] Store $U$ in the two-qubit program
$$
|P_U\rangle=(I\otimes U)|\phi^+\rangle.
$$
Apply the teleportation Bell measurement to the input and the first program qubit. On the $\phi^+$ outcome, which has probability $1/4$ independently of $U$ and $|\alpha\rangle$, the remaining qubit is exactly $U|\alpha\rangle$. Accept that heralded outcome and declare failure otherwise. This gives a probabilistic universal programmable gate with constant success probability $p=1/4$.
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