  Let $G$ be a multigraph. A subset $F$ of $E(G)$ is an edge cover of $G$ if
every vertex of $G$ is incident to an edge of $F$. The cover index, $\xi(G)$,
is the largest number of edge covers into which the edges of $G$ can be
partitioned. Clearly $\xi(G) \le \delta(G)$, the minimum degree of $G$. For
$U\subseteq V(G)$, denote by $E^+(U)$ the set of edges incident to a vertex of
$U$. When $|U|$ is odd, to cover all the vertices of $U$, any edge cover needs
to contain at least $(|U|+1)/2$ edges from $E^+(U)$, indicating $ \xi(G) \le
|E^+(U)|/ (|U|+1)/2$. Let $\rho_c(G)$, the co-density of $G$, be defined as the
minimum of $|E^+(U)|/((|U|+1)/2)$ ranging over all $U\subseteq V(G)$ with $|U|
$ odd and at least 3. Then $\rho_c(G)$ provides another upper bound on
$\xi(G)$. Thus $\xi(G) \le \min\{\delta(G), \lfloor \rho_c(G) \rfloor \}$. For
a lower bound on $\xi(G)$, in 1967, Gupta conjectured that $\xi(G) \ge
\min\{\delta(G)-1, \lfloor \rho_c(G) \rfloor \}$. Gupta showed that the
conjecture is true when $G$ is simple, and Cao et al. verified this conjecture
when $\rho_c(G)$ is not an integer. In this note, we confirm the conjecture
when the maximum multiplicity of $G$ is at most two or $ \min\{\delta(G)-1,
\lfloor \rho_c(G) \rfloor \} \le 6$.
