The volume ratio of a convex body $K\subset \mathbb{R}^{n}$ is $v_r(K) = \inf_{\mathcal{E}\subset K} \left(\frac{Vol(K)}{Vol(\mathcal{E})}\right)^{1/n}$ where the infimum run over ellipsoids included in K

By using John's Theorem, Keith Ball showed that the cube has largest volume ratio among symmetric convex bodies (of order $\sqrt{n}$).

Define dually $v_r^\circ(K) = \inf_{K\subset \mathcal{E}} \left(\frac{Vol(\mathcal{E})}{Vol(K)}\right)^{1/n}$ where the infimum run over ellipsoids containing K. By duality and using Bourgain-Milman theorem which says that the quantity $Vol(K)Vol(K^\circ)^{1/n}$ is upper and lower bounded on the class of convex bodies by two absolute constants, we see that $v_r^\circ(K)$ is up to constant maximised by the cube among symmetric convex bodies (I suspect it is indeed the maximum).

My question is : is there a quantitative stability result in terms of the number of faces ? That is, if $K$ has $k$ faces is there a result looking like

$$v_r(K) \geq c_0 \frac{\sqrt{n}}{1+\log(k/n)} $$

Or dually

$$v_r^\circ(K) \leq c_0 (1+\log(k/n)) $$ something like that ?