Before revision:  More generally, when it comes to scalarizing a multivariate risk measure with the same scalarization weight at each time (as this is the case in this paper as the risk measures are expressed in the cash asset), one cannot expect these scalar risk measures to be time consistent. This aligns with recent research on scalarizations of multivariate problems as in Karnam et al. (2017) and Kovacova and Rudloff (2018). Surprisingly, we will show that a weaker notion of time consistency can be defined, which corresponds to the usual scalar time consistency but under any fixed consistent pricing process.
After revision:  We will show that a weaker notion of time consistency can be defined, which corresponds to the usual scalar time consistency but under any fixed consistent pricing process.