Inverse probability weights with time-varying treatments $A_t$ and confounders $L_t$ are defined as the inverse probability of being treated at time $t$ conditional on past treatment and covariate history: $$w_i=\prod_{t=1}^{T}\frac{1}{P(A_t=A_{i,t}|\overline{A}_{t-1}=\overline{A}_{i,t-1},\overline{L}_{t}=\overline{l}_{i,t})}$$
I'm having a hard time understanding why this formula provides sequential ignorability under no unobserved confounders. Indeed, according to Robins, Hernan and people that built this estimator (see e.g., Hernan et al., 2009), this leads to simulate
a pseudo-population in which treatment $A_t$ does not depend on prior covariate history $L_{t−1}$, i.e. a pseudo-population in which the arrow from $L_t$ to $A_t$ does not exist.
What I don't understand here is why does, in the corresponding DAG, the arrow from $A_t$ to $A_{t-1}$ is still present. In other words, while conditioning on past confounder at each time point does remove the arrow from past confounders to current treatment, conditioning on past treatments at each time point does not remove the causal link between the two variables. Why is that?