We characterize the U.S. interbank liquidity risk network based on a supervisory dataset, using a scenario-based quantile network connectedness approach. In terms of methodology, we consider a quantile vector autoregressive model with unobserved heterogeneity and propose a Bayesian nuclear norm estimation method. A common factor structure is employed to deal with unobserved heterogeneity that may exhibit endogeneity within the network. Then we develop a scenario-based quantile network connectedness framework by accommodating various economic scenarios, through a scenario-based moving average expression of the model where forecast error variance decomposition under a future pre-specified scenario is derived. The methodology is used to study the quantile-dependent liquidity risk network among large U.S. bank holding companies. The estimated quantile liquidity risk network connectedness measures could be useful for bank supervision and financial stability monitoring by providing leading indicators of the system-wide liquidity risk connectedness not only at the median but also at the tails or even under a pre-specified scenario. The measures also help identify systemically important banks and vulnerable banks in the liquidity risk transmission of the U.S. banking system.