= Solution
Write $r=\theta_A+\theta_B$, with $\theta_A\ge0$, $\theta_B>0$ and a finite $\tau\ge0$. The <competing risks model with transient surgical mortality> has <survival function>
$$
S(t)=\exp\{-\theta_Bt-\theta_A\min(t,\tau)\}.
$$
Integrating the disease <cause-specific hazard> against this <survival function> gives
$$
\boxed{F_B(t)=\begin{cases}\dfrac{\theta_B}{r}(1-e^{-rt}),&0\le t\le\tau,\\[4pt]\dfrac{\theta_B}{r}(1-e^{-r\tau})+e^{-r\tau}\bigl(1-e^{-\theta_B(t-\tau)}\bigr),&t>\tau.\end{cases}}
$$
The first term accounts for disease deaths while both causes act; the second includes survivors of that period who then face only disease mortality. The <probability> of eventually dying from surgery is
$$
\boxed{P(J=A)=F_A(\infty)=\int_0^\tau\theta_Ae^{-ru}\,du=\frac{\theta_A}{r}(1-e^{-r\tau}).}
$$
Taking the limit in $F_B$ gives $F_B(\infty)=\theta_B(1-e^{-r\tau})/r+e^{-r\tau}$, and therefore
$$
\boxed{F_A(\infty)+F_B(\infty)=1,\qquad S(\infty)=0.}
$$
The ultimate-death conclusion uses the positive continuing disease hazard. If $\theta_B=0$, the surviving fraction $e^{-\theta_A\tau}$ would instead live indefinitely in this model; the asserted conclusion is not valid in that boundary case.
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