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

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?