= Solution
Write $\mu_i=\alpha+\beta z_i+\gamma^{\mathsf T}x_i$ and let $f_i(y)=\sigma^{-1}\phi((y-\mu_i)/\sigma)$ be the <normal distribution> density. Set $q_j(y)=\operatorname{expit}(\eta_j+\theta y)$, where $q_j$ is the probability that attempt $j$ is unsuccessful conditional on the preceding attempts being unsuccessful. This is a <sequential response selection model>. For first-attempt success, the observed outcome and attempt count contribute
$$
\boxed{L_i=f_i(y_i)[1-q_1(y_i)]
=\frac{\phi((y_i-\mu_i)/\sigma)}{\sigma[1+e^{\eta_1+\theta y_i}]}.}
$$
This <likelihood> conditions on baseline and assignment; the displayed response probabilities do not use $d_i$. No factor for random allocation is needed when conditioning on its realized value.
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