= Solution
An ideal projective measurement yielding the antisymmetric state prepares that state at $t_1$. Starting from it, the <probability> of reaching the symmetric state after an additional interval $s$ is again $\sin^2((E_2-E_1)s/(2\hbar))$. At the full recurrence interval found above,
$$
\boxed{P\bigl(+\text{ at }t_1+\tau\mid-\text{ at }t_1\bigr)=\sin^2\pi=0.}
$$
The full-period propagator is proportional to the identity on the two-dimensional subspace, so it returns the antisymmetric state to itself, not to its orthogonal partner. The result assumes the stated time-independent evolution continues after the measurement.
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