  The refined Kaneko-Zagier conjecture claims that the algebras spanned by two
kinds of "completed" finite multiple zeta values, called $\hat{A}$- and
$\hat{S}$-MZVs, are isomorphic. Recently, Komori defined $\hat{S}$-MZVs of
general integer (i.e., not necessarily positive) indices, extending the
existing definition for positive indices. In view of the refined Kaneko-Zagier
conjecture, Komori's work suggests that these extended values are closely
connected to $\hat{A}$-MZVs of general indices, which can be defined in an
obvious way. In this paper, we show that the generalization of the refined
Kaneko-Zagier conjecture for general integer indices is actually deduced from
the conjecture for positive indices. The key ingredient is an inductive formula
for $\hat{A}$-MZVs or $\hat{S}$-MZVs of indices which contain at least one
non-positive entry.
