 Solution

Yes. Store $U$ in the two-qubit programApply the teleportation Bell measurement to the input and the first program qubit. On the ${\varphi }^{+}$ outcome, which has probability $1/4$ independently of $U$ and $|\alpha ⟩$, the remaining qubit is exactly $U|\alpha ⟩$. Accept that heralded outcome and declare failure otherwise. This gives a probabilistic universal programmable gate with constant success probability $p=1/4$.