The symmetric Pisier estimate for the $ell$-functional

Let $K\subset\mathbb R^n$ be an origin-symmetric convex body, let $GL_n$ denote the group of invertible linear transformations of $\mathbb R^n$, and let $\ell(K)$ be the product functional discussed in the paper. Symmetric Pisier estimate. There exists a universal constant $C>0$ such that, for every origin-symmetric convex body $K\subset\mathbb R^n$, one can find $T\in GL_n$ satisfying

$$\ell(TK)\leq C\sqrt{\log n}.$$

The estimate would improve the general affine bound of order $\log n$ for origin-symmetric convex bodies; the supplied text presents it as believed rather than established, so its resolution remains open.