  We analyse the complexity of approximate counting constraint satisfactions
problems $\mathrm{\#CSP}(\mathcal{F})$, where $\mathcal{F}$ is a set of
nonnegative rational-valued functions of Boolean variables. A complete
classification is known in the conservative case, where $\mathcal{F}$ is
assumed to contain arbitrary unary functions. We strengthen this result by
fixing any permissive strictly increasing unary function and any permissive
strictly decreasing unary function, and adding only those to $\mathcal{F}$:
this is weak conservativity. The resulting classification is employed to
characterise the complexity of a wide range of two-spin problems, fully
classifying the ferromagnetic case. In a further weakening of conservativity,
we also consider what happens if only the pinning functions are assumed to be
in $\mathcal{F}$ (instead of the two permissive unaries). We show that any set
of functions for which pinning is not sufficient to recover the two kinds of
permissive unaries must either have a very simple range, or must satisfy a
certain monotonicity condition. We exhibit a non-trivial example of a set of
functions satisfying the monotonicity condition.
