  The Graded Classification Conjecture states that the pointed
$K_0^{\operatorname{gr}}$-group is a complete invariant of the Leavitt path
algebras of finite graphs when these algebras are considered with their natural
grading by $\mathbb Z.$ The strong version of this conjecture states that the
functor $K_0^{\operatorname{gr}}$ is full and faithful when considered on the
category of Leavitt path algebras of finite graphs and their graded
homomorphisms modulo conjugations by invertible elements of the zero
components. We show that the functor $K_0^{\operatorname{gr}}$ is full for the
unital Leavitt path algebras of countable graphs and that it is faithful
(modulo specified conjugations) only in a certain weaker sense.
