\overline{L}_{s,i}, T_i\ge t_s, C_i \ge t_s)}\cdot \frac{I(A_i = a)}{\hat{\text{Pr}}(A_i=a|L_{0,i})}\cdot\\
&\,\cdot\frac{1}
{\bm\prod_{s: 0\le s \le t} \left[1- \hat{\lambda}_C(s|T_{i}>s, \overline{M}_{\lfloor s\rfloor ,i}, \overline{L}_{\lfloor s\rfloor,i},L_{0,i},A_i=a)\right]}\end{aligned}$$ and $\lambda_C(.)$ denoting the cause-specific hazard function of the time-to-censoring. Here, $\bm\prod_{s} x_s$ is defined as a product limit. With the above weight $W_i(\lfloor t \rfloor,a,a^*)$, the additional component that accounts for the informative censoring can make the overall weight become unstable (e.g. when the censoring hazard is close to 1 in some strata). To overcome this, one can then use stabilized (censoring) weights which incorporate a numerator defined in the same way as the denominator but adjusting only for the exposure, that is: $$\frac{\bm\prod_{s: 0\le s \le t} \left[1- \hat{\lambda}_C(s|A_i=a)\right]}
{\bm\prod_{s: 0\le s \le t} \left[1- \hat{\lambda}_C(s|T_{i}>s, \overline{M}_{\lfloor s\rfloor ,i}, \overline{L}_{\lfloor s\rfloor,i},L_{0,i},A_i=a)\right]}$$ One then needs to postulate two models for the censoring hazard at time $t$, with one conditioning on exposure and the other conditioning on exposure, baseline covariates and the history of the longitudinal mediator and confounders up to time $t$, where only the latter model needs to be correct. For instance, $\lambda_C(t|A_i = a_i)=\lambda'_{0C}(t)e^{\eta a_i}$ and $$\begin{aligned}
 \lambda_C(t|T_{i}>t, \overline{M}_{\lfloor s\rfloor,i}=\overline{m}_{\lfloor s\rfloor,i}, \overline{L}_{\lfloor s\rfloor,i}=\overline{l}_{\lfloor s\rfloor,i},L_{0,i}=l_{0,i},A_i=a_i)=\lambda_{0C}(t)e^{\theta_0a_i + \theta_1l_{0,i} + \theta_2'\overline{m}_{\lfloor s\rfloor ,i} + \theta_3'\overline{l}_{\lfloor s\rfloor,i}}\end{aligned}$$ As a 