= Solution
Here
$$
C^*=C_X+b^2\phi_{K+1}\otimes\phi_{K+1}.
$$
It has the same eigenfunctions as $C_X$, but the $(K+1)$st eigenvalue is raised from $\lambda_{K+1}$ to $\lambda_{K+1}+b^2$. A test based on the first $K$ eigenfunctions of $C_X$ is blind to this alternative because $\delta=b\phi_{K+1}$ is orthogonal to all of them. If the raised eigenvalue overtakes at least $\lambda_K$, then $\phi_{K+1}$ enters the first $K$ eigenfunctions of $C^*$ and the mean shift acquires a retained nonzero coordinate. The $C^*$-based test can therefore detect alternatives that the original FPCA truncation misses.
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