  Let $G=(V,E)$ be a multigraph. The {\em cover index} $\xi(G)$ of $G$ is the
greatest integer $k$ for which there is a coloring of $E$ with $k$ colors such
that each vertex of $G$ is incident with at least one edge of each color. Let
$\delta(G)$ be the minimum degree of $G$ and let $\Phi(G)$ be the {\em
co-density} of $G$, defined by \[\Phi(G)=\min
\Big\{\frac{2|E^+(U)|}{|U|+1}:\,\, U \subseteq V, \,\, |U|\ge 3 \hskip 2mm {\rm
and \hskip 2mm odd} \Big\},\] where $E^+(U)$ is the set of all edges of $G$
with at least one end in $U$. It is easy to see that $\xi(G) \le
\min\{\delta(G), \lfloor \Phi(G) \rfloor\}$. In 1978 Gupta proposed the
following co-density conjecture: Every multigraph $G$ satisfies $\xi(G)\ge
\min\{\delta(G)-1, \, \lfloor \Phi(G) \rfloor\}$, which is the dual version of
the Goldberg-Seymour conjecture on edge-colorings of multigraphs. In this note
we prove that $\xi(G)\ge \min\{\delta(G)-1, \, \lfloor \Phi(G) \rfloor\}$ if
$\Phi(G)$ is not integral and $\xi(G)\ge \min\{\delta(G)-2, \, \lfloor \Phi(G)
\rfloor-1\}$ otherwise. We also show that this co-density conjecture implies
another conjecture concerning cover index made by Gupta in 1967.
