= Solution
After local phases, take $\alpha,\beta\ge0$. Alice measures with Kraus operators $M_1=\operatorname{diag}(\alpha,\beta)$ and $M_2=\operatorname{diag}(\beta,\alpha)$, which satisfy $M_1^\dagger M_1+M_2^\dagger M_2=I$. Outcome 1 directly gives $\alpha|00\rangle+\beta|11\rangle$. Outcome 2 gives $\beta|00\rangle+\alpha|11\rangle$; Alice tells Bob the outcome and both apply $X$, swapping the two coefficients. Thus the conversion succeeds with certainty, in agreement with Nielsen majorization.
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