= Solution
Let $q_{rs}$ be the <transition intensity> from state $r$ to $s$, and let $Q$ be the <transition intensity matrix>, with $q_{rr}=-\sum_{s\ne r}q_{rs}$. For a <continuous-time multi-state model> with the <time-homogeneous Markov property>, the <transition probability matrix> is $P(u)=e^{uQ}$, with entry $p_{rs}(u)$. Condition on each recorded initial state. Panel visits contribute <transition probabilities>; an exact entry into the absorbing death state contributes a <statistical probability density>
$$
g_{r3}(u)=\sum_{s=1}^2p_{rs}(u)q_{s3}.
$$
This <mixed panel and exact-death likelihood> sums over the living state just before death. It accounts for survival until the event; replacing its final factor by $p_{r3}(u)$ would count deaths throughout the interval.
Using the actual visit times recovered from the PDF, the three individual <likelihood> contributions are
$$
\begin{aligned}
L_7={}&p_{12}(2.473380)p_{22}(3.708143)p_{22}(0.114158)
p_{22}(0.811791)p_{22}(0.359291)p_{22}(0.457878)g_{23}(0.065913),\\
L_8={}&p_{11}(3.261286)p_{11}(1.231073),\\
L_9={}&p_{11}(1.289561)p_{11}(2.694442)p_{11}(0.450082)
p_{12}(4.338740)p_{22}(0.273262).
\end{aligned}
$$
Here $g_{23}$ means $g_{r3}$ with $r=2$, not a <transition probability>. Subjects 8 and 9 supply no event-density factor after their last panel observation. Assume independent subjects and noninformative examination and <censoring> times; conditional on their observation schedule, its distribution supplies no additional <statistical parameter>-dependent factor. The patients' <covariates> can be incorporated by using their own $Q_i$ in these same expressions.
For the progressive structure used in the subsequent output, put $a=q_{12}$, $b=q_{13}$, $c=q_{23}$ and $\lambda=a+b$. The <progressive illness-death model> permits no recovery, so
$$
p_{11}(u)=e^{-\lambda u},\qquad p_{22}(u)=e^{-cu},\qquad
p_{12}(u)=\frac{a}{\lambda-c}(e^{-cu}-e^{-\lambda u}),\qquad
g_{23}(u)=ce^{-cu}.
$$
When $\lambda=c$, the continuous limit is $p_{12}(u)=au e^{-cu}$. Thus the <likelihood> contributions simplify to
$$
\boxed{\begin{aligned}
L_7&=p_{12}(2.473380)c\,e^{-c(7.990554-2.473380)},\\
L_8&=e^{-\lambda(4.492359)},\\
L_9&=e^{-\lambda(4.434085)}p_{12}(4.338740)e^{-c(0.273262)}.
\end{aligned}}
$$
Intermediate unobserved disease transitions remain integrated into each panel <transition probability>.
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