  Given a monotone Lagrangian $L$ in a compact symplectic manifold $X$, we
construct a commutative diagram relating the closed-open string map $CO_\lambda
: QH^*(X) \to HH^*(\mathcal{F}(X)_\lambda)$ to a variant of the length-zero
closed-open map on $L$ incorporating $\mathbf{k}[H_1(L; \mathbb{Z})]$
coefficients, denoted $CO^0_\mathbf{L}$. The former is categorically important
but very difficult to compute, whilst the latter is geometrically natural and
amenable to calculation. We further show that, after a suitable completion,
injectivity of $CO^0_\mathbf{L}$ implies injectivity of $CO_\lambda$. Via
Sheridan's version of Abouzaid's generation criterion, this gives a powerful
tool for proving split-generation of the Fukaya category. We illustrate this by
showing that the real part of a monotone toric manifold (of minimal Chern
number at least 2) split-generates the Fukaya category in characteristic 2. We
also give a short new proof (modulo foundational assumptions in the
non-monotone case) that the Fukaya category of an arbitrary compact toric
manifold is split-generated by toric fibres.
