In our earlier paper [{\bf 10}], for a foundation $*-$semigroup $S$ with
an
identity and with a Borel-measurable weight function $w\leq 1$, we proved
that on the unit ball of ${\cal P} (S,w)$, the cone of $w-$bounded continuous
positive definite functions on $S$, the weak topology coincides with the
compact open topology. In the present paper, through some $C^*-$algebras
techniques, we shall extend this result to the unit ball of the
Fourier-Stieltjes algebra ${\cal F} (S,w)$
of a foundation semigroup $S$ with a
Borel measurable weight function $w$. Indeed, we shall establish our
conjecture in [{\bf 10}] even in the more general setting of the
Fourier-Stieltjes algebra ${\cal F}(S,w)$
for any Borel measurable weight function
$w$. It should be noted that the family of foundation semigroups is quite
extensive, for which locally compact groups and discrete semigroups
are elementary examples. For further examples we refer to Appendix B of
[{\bf 13}].