= Solution
Couple the <Wald test> statistics as $W_i=Z_i+\delta_i/s_i$, where the joint centered <bivariate normal distribution> of $(Z_1,Z_2)$ is fixed. Increasing either $\delta_i$ enlarges, pointwise in this coupling, the event $\{W_1>c\text{ or }W_2>c\}$. Its probability is consequently nondecreasing in each treatment effect.
If both <null hypotheses> hold, then $\delta_1,\delta_2\le0$, so this probability is maximal at the joint boundary $(0,0)$. To check the <familywise error rate> across all configurations, suppose only hypothesis $i$ is true. A <type I error> then occurs only if $W_i>c$, with probability at most $1-\Phi(c)$, where $\Phi$ is the <standard normal distribution function>. This is no larger than the any-rejection probability at $(0,0)$, since that joint event contains $\{Z_i>c\}$. If neither <null hypothesis> is true, the <familywise error rate> is zero. Hence \b[the <least-favourable null configuration> for <type I error>] is
$$
\boxed{\delta_1=\delta_2=0,\qquad\operatorname{FWER}_{\max}=1-\Phi_2(c,c;\rho),}
$$
where $\Phi_2$ is the centered <bivariate normal distribution> function with <correlation coefficient> $\rho$.
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