= Solution
Let
$$
\overline C=\frac12(C_X+C_{X^*}),
\qquad
\overline C\phi_k=\lambda_k\phi_k,
$$
and estimate it by the average of the two within-sample <empirical covariance operator>[empirical covariance operators]. Let $(\widehat\lambda_k,\widehat\phi_k)$ be its leading empirical eigenpairs and let $\overline X_n,\overline X_n^*$ be the sample means. Use the <Two-sample FPCA mean statistic>
$$
\boxed{
T_{n,K}=\frac n2\sum_{k=1}^K
\frac{\langle\overline X_n-\overline X_n^*,\widehat\phi_k\rangle^2}
{\widehat\lambda_k}}.
$$
Under $H_0$, the <Hilbert-space central limit theorem> gives
$$
\sqrt{\frac n2}(\overline X_n-\overline X_n^*)
\xrightarrow dG,
$$
where $G$ is a centered <Gaussian random element> with covariance $\overline C$. Distinct eigenvalues give consistent empirical eigenpairs, up to signs, and the standardized leading scores are independent standard normal variables. Hence
$$
T_{n,K}\xrightarrow d\chi_K^2.
$$
Rejecting above the $(1-\alpha)$ quantile gives an asymptotic level-$\alpha$ test.
Under a fixed alternative $\delta=\mu-\mu^*$,
$$
\frac{T_{n,K}}n\xrightarrow p
\frac12\sum_{k=1}^K\frac{\langle\delta,\phi_k\rangle^2}{\lambda_k}.
$$
The test is consistent whenever one retained projection is nonzero. Alternatives orthogonal to the first $K$ eigenfunctions are invisible at fixed $K$. Under local alternatives $\delta=h/\sqrt n$, the limit is noncentral chi-squared with noncentrality $\frac12\sum_{k\leq K}\langle h,\phi_k\rangle^2/\lambda_k$.
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