 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 ${\sin }^2((E_2-E_1)s/(2\hslash ))$. At the full recurrence interval found above,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.