 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 ${\delta }_1,{\delta }_2\le 0$, 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 the least-favourable null configuration for type I error iswhere ${\Phi }_2$ is the centered bivariate normal distribution function with correlation coefficient $\rho$.