fair chore allocation with qualification

Background

There are $$n$$ indivisible heterogenous bads (chores).
There are $$m$$ agents.
The subjective utility functions of the agents are additive and identical. e.g $$\forall X,i,j\ V_i(X) = V_j(X)$$.
The agents have different entitlements over the chores.
Finally, there is an additional constraint: there are chores that some agents cannot be assigned to perform (i.e the qualification constraint)

I believe that due to the qualification constraint some standard fairness criteria, like $$EF1$$, may not exist. This is why I am also asking for a fairness criterion.