Pisier's Grothendieck-pair conjecture with the bounded approximation property

Let $\mathcal{E}=(E_n)_{n\geqslant 1}$ be a sequence of finite-dimensional Banach spaces with $\dim E_n=n$, and let $F$ be a Banach space with the bounded approximation property. A pair $(\mathcal{E},F)$ is a Grothendieck pair if there is a constant $C>0$ such that

$$\|A\otimes \operatorname{id}_{F}\|_{E_n\check{\otimes}F\to E_n^*\hat{\otimes}F}\leq C\|A\|_{E_n\to E_n^*}$$

for every $n\in\mathbb{N}$. Pisier's Grothendieck-pair conjecture. If $(\mathcal{E},F)$ is a Grothendieck pair, then either $\dim F<\infty$ or $(\mathcal{E},\ell_2)$ is also a Grothendieck pair. This asks for a classification of Grothendieck pairs; it is known in the paper when $F$ is a GL-space, but remains open for general Banach spaces with the bounded approximation property.