Proportionality (PROP) is one of the simplest and most intuitive fairness criteria used for allocating items among agents with additive utilities. However, when the items are indivisible, ensuring PROP becomes unattainable, leading to increased focus on its relaxations. In this paper, we focus on the relaxation of proportionality up to any item (PROPX), where proportionality is satisfied if an arbitrary item is removed from every agent's allocation. We show that PROPX is an appealing fairness notion for the allocation of indivisible chores, which approximately implies some share-based notions, such as maximin share (MMS) and AnyPrice share (APS). We further provide a comprehensive understanding of PROPX allocations, regarding the computation, approximation, and compatibility with efficiency. On top of these, we extend the study to scenarios where agents do not share equal liability towards the chores, and approximate PROPX allocations using partial information about agents' utilities. •We design algorithms to compute PROPX allocations for indivisible chores when the agents have asymmetric weights.•We design algorithms, which only access the agents' ordinal preferences, to compute approximate PROPX allocations.•We investigate the compatibility of PROPX and Pareto optimality in the general and special cases.