Recall that a Banach space $X$ has the approximation property (AP for short ) if for every compact subset $K$ of $X$ and every $\varepsilon > 0$, there exists a finite rank operator $S$ on $X$ such that $\|Sx-x\|<\varepsilon$ for all $x\in K$. If, in addition, there exists $\lambda \geq 1$ such that we can always choose a finite rank operator $S$ on $X$ with $\|S\|\leq \lambda$, then we say that $X$ has the $\lambda$-bounded approximation property ($\lambda$-BAP). If a Banach space $X$ has the $\lambda$-BAP for some $\lambda$, we say that $X$ has the BAP.
Let $X,Y$ be Banach spaces. We denote by $\mathcal{L}_{w^{*}}(X^{*},Y),\mathcal{K}_{w^{*}}(X^{*},Y),\mathcal{F}_{w^{*}}(X^{*},Y)$ by the classes of all $w^{*}$-to-$w$ continuous operators, $w^{*}$-to-$w$ continuous compact operators and $w^{*}$-to-$w$ continuous finite rank operators from $X^{*}$ to $Y$, respectively. In 1955, A. Grothendieck proved that a Banach space $X$ has the AP if and only if $\mathcal{F}_{w^{*}}(X^{*},Y)$ is dense on $\mathcal{K}_{w^{*}}(X^{*},Y)$ in the operator norm topology for all Banach spaces $Y$.
I am thinking about a quantitative version of this well-known result. Let me fix some notations. If $A$ and $B$ are two nonempty subsets of a Banach space $X$, we set $$d(A,B)=\inf\{\|a-b\|:a\in A,b\in B\},$$$$\widehat{d}(A,B)=\sup\{d(a,B):a\in A\}.$$ Let $A$ be a bounded subset of a Banach space $X$. The Hausdorff measure of non-compactness of $A$ is defined by $\chi(A)=\inf\{\widehat{d}(A,F):F\subseteq X$ finite subsets$\}$. Then $\chi(A)=0$ if and only if $A$ is relatively norm compact. For an operator $T: X\rightarrow Y$, $\chi(T)$ will denote $\chi(TB_{X})$. I have proved the following result:
Proposition. If a Banach space $X$ has the BAP, then there exists a constant $\mu>0$ such that $$d(T,\mathcal{F}_{w^{*}}(X^{*},Y))\leq \mu \chi(T^{*})$$ for all Banach spaces $Y$ and all $T\in \mathcal{L}_{w^{*}}(X^{*},Y)$.
Question. Is the converse of Proposition true?
