  We show that any pointed, preordered module map
$\mathfrak{BF}_{\mathrm{gr}}(E) \to \mathfrak{BF}_{\mathrm{gr}}(F)$ between
Bowen-Franks modules of finite graphs can be lifted to a unital, graded,
diagonal preserving $\ast$-homomorphism $L_\ell(E) \to L_\ell(F)$ between the
corresponding Leavitt path algebras over any commutative unital ring with
involution $\ell$. Specializing to the case when $\ell$ is a field, we
establish the fullness part of Hazrat's conjecture about the functor from
Leavitt path $\ell$-algebras of finite graphs to preordered modules with order
unit that maps $L_\ell(E)$ to its graded Grothendieck group. Our construction
of lifts is of combinatorial nature; we characterize the maps arising from this
construction as the scalar extensions along $\ell$ of unital, graded
$\ast$-homomorphisms $L_{\mathbb Z}(E) \to L_{\mathbb Z}(F)$ that preserve a
sub-$\ast$-semiring introduced here.
