 Solution

After local phases, take $\alpha ,\beta \ge 0$. Alice measures with Kraus operators $M_1=\mathrm{diag}(\alpha ,\beta )$ and $M_2=\mathrm{diag}(\beta ,\alpha )$, which satisfy $M_1^{†}{M}_1+M_2^{†}{M}_2=I$. Outcome 1 directly gives $\alpha |00⟩+\beta |11⟩$. Outcome 2 gives $\beta |00⟩+\alpha |11⟩$; Alice tells Bob the outcome and both apply $X$, swapping the two coefficients. Thus the conversion succeeds with certainty, in agreement with Nielsen majorization.