= Solution
Four unsuccessful attempts have conditional probability $\prod_{j=1}^4q_j(y)$. The outcome is then unobserved, so the <observed-data likelihood> integrates over it:
$$
\boxed{L_i=\int_{-\infty}^{\infty}
\frac1\sigma\phi\left(\frac{y-\mu_i}{\sigma}\right)
\prod_{j=1}^{4}\frac{e^{\eta_j+\theta y}}{1+e^{\eta_j+\theta y}}\,dy.}
$$
This is a probability, whereas a responder contributes a joint density in the observed outcome. The <sequential response selection model> keeps outcome-dependent nonresponse in the <likelihood>; evaluating the product at an imputed mean would not be this likelihood.
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