  For a compact space $K$, the Banach space $C(K)$ is said to have the
$\ell_1$-Grothendieck property if every weak* convergent sequence
$\big\langle\mu_n\colon\ n\in\omega\big\rangle$ of functionals on $C(K)$ such
that $\mu_n\in\ell_1(K)$ for every $n\in\omega$, is weakly convergent. Thus,
the $\ell_1$-Grothendieck property is a weakening of the standard Grothendieck
property for Banach spaces of continuous functions. We observe that $C(K)$ has
the $\ell_1$-Grothendieck property if and only if there does not exist any
sequence of functionals $\big\langle\mu_n\colon\ n\in\omega\big\rangle$ on
$C(K)$, with $\mu_n\in\ell_1(K)$ for every $n\in\omega$, satisfying the
conclusion of the classical Josefson--Nissenzweig theorem. We construct an
example of a separable compact space $K$ such that $C(K)$ has the
$\ell_1$-Grothendieck property but it does not have the Grothendieck property.
We also show that for many classical consistent examples of Efimov spaces $K$
their Banach spaces $C(K)$ do not have the $\ell_1$-Grothendieck property.
