Shared care includes noticing what needs doing
A shared task involves more than the visible action. Someone must notice the empty cupboard, decide what is needed and remember when to replace it. If one person always manages those steps, asking them to assign every job still leaves much of the responsibility with them.
Go deeper
Look at the whole responsibility
Buying household supplies is the visible part of a larger task. Before the purchase, someone checks what is running low. Afterward, someone may put things away and track what remains missing.
Owning a task can include noticing, planning and following through. Merely waiting for instructions can leave the coordinating work with someone else.
A clear arrangement might give one person responsibility for keeping a particular set of supplies available, including checking and replenishing them. The arrangement can be reviewed when schedules change.
The aim is not to assume what everyone can manage. It is to make the less visible work discussable, so a shared responsibility does not quietly become one person's permanent role as organiser.