 Solution

First use the allowed one-query algorithm to learn $p=x_1\oplus x_2$ exactly. This restricted two-index Boolean quantum oracle is realized by relabelling indices $1,2$ in a single query to the original input. Measure its output and choose the second query classically.

If $p=0$, the first two bits agree and their common value is the majority, irrespective of $x_3$. Query $x_1$ and output it. If $p=1$, the first two bits cancel in the vote, so query $x_3$ and output it. ThusThe exact two-query majority algorithm uses one parity query and one ordinary bit query on every branch, and is correct on all eight inputs. Together with the lower bound this proves $Q_E(\mathrm{MAJ})=2$.