Are there "small cardinal models" of a Banach space, whose dual space is in Stegall`s class, but it is not $weak^{*}$-fragmentable, or large cardinals are a must?

Abstract

It is well-known that if $Y$ is a Banach space the $weak^{*}$-fragmentability of its dual space by some metric $\rho$ implies that $Y^{*}$ belongs to the Stegall class -- the former for shortly $\mathcal{W^*}F$, being the latter $\mathcal{S}$ and hence $Y$ is weak Asplund -- call it $\mathcal{WA}$ . It has been proved by O. Kalenda and Kunen that existence of a measurable cardinal implies (it is consistent) that, for instance in the construction of Kalenda Compacts - this space is in the Stegall`s class iff both inclusions of classes are strictly proper. The same authors made following question, is there a model of ZFC, in which the inclusion of $\mathcal{WA}$ in $\mathcal{S}$ actually is equality. Obviously, because of their result the existence of large cardinals ( supercompacts, strongly-compacts, strong cardinals, huge cardinals, Vop\v{e}nka principle) would be a models of the proper inclusion of the above - mentioned classes, being with more consistency power even, see Thomas Jech [TJ03] and Saharon Shelah [SSH17].