= Solution
Let $r$ be the input <Schmidt rank>, so $\lambda_j=0$ for $j>r$ and $\sum_{j=1}^r\lambda_j=1$. If $r=d$, the output rank is already at most $r$. Otherwise, <Nielsen's pure-state conversion theorem> and <majorization> at $k=r$ give
$$
1=\sum_{j=1}^r\lambda_j\leq\sum_{j=1}^r\mu_j\leq\sum_{j=1}^d\mu_j=1.
$$
Thus all output coefficients beyond $r$ vanish, proving
$$
\boxed{\operatorname{Schmidt\ rank}(|\phi\rangle)\leq\operatorname{Schmidt\ rank}(|\psi\rangle).}
$$
This is <monotonicity of Schmidt rank under LOCC>. It also holds separately in any nonzero postselected branch: represent the input amplitudes by a matrix $C$; a local branch maps it to $ACB^T$, whose rank cannot exceed the rank of $C$. The deterministic result requested here follows already from the majorization criterion.
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