= Solution
Let $p_{rs}(t)=P(X(u+t)=s\mid X(u)=r)$ and $P(t)=e^{Qt}$, where time homogeneity removes dependence on $u$. A recorded state at the next clinic visit contributes a <transition probability>; an exactly observed death contributes a <statistical probability density>, not the <probability> of being dead at that time. If the last recorded living state is $r$, the <mixed panel and exact-death likelihood> factor after an interval $t$ is
$$
g_{r3}(t)=\sum_{j=1}^2p_{rj}(t)q_{j3}=\frac{d}{dt}p_{r3}(t).
$$
This sums over the unobserved living state immediately before death.
Conditioning on the recorded initial states, the contribution of the three displayed patient histories is
$$
\boxed{L(Q)=p_{11}(8.5)\,p_{11}(26.3)\,p_{22}(12.6)\left[p_{21}(34.6)q_{13}+p_{22}(34.6)q_{23}\right].}
$$
In this irreversible <illness-death model>, $p_{21}=0$, $p_{11}(t)=e^{-(a+b)t}$ and $p_{22}(t)=e^{-ct}$, simplifying it to
$$
\boxed{L(Q)=c\exp\{-34.8(a+b)-47.2c\}.}
$$
The factor $c$ is essential: the death time is known exactly. Replacing the final <statistical probability density> by $p_{23}(34.6)$ would instead model interval observation of death and give a different <likelihood>.
The assumptions are independent patient histories with common rates; the Markov property; constant rates over calendar/follow-up time in this model; the stated absence of recovery and absorption at death; accurate state labels and death times; and an observation/follow-up mechanism that is noninformative for the latent process given the observed history. Clinic dates are conditioned on. The displayed living endpoints contribute only the shown observations, with noninformative <right censoring> if they are follow-up endpoints. Initial state <probabilities> are omitted by conditioning on them. Progression between visits can be unobserved, which is precisely why <panel-observed multi-state likelihood> uses the <matrix exponential> rather than assuming a transition occurs at a visit.
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