By translating and subtracting a linear function assume that $x=p=0$. Assume by way of contradiction that we can find $h_{k}\rightarrow 0$ and some $\delta>0$ such that

$$d_{n-1}(h_{k})>\delta h_{k}^{1/2}$$

for all $k$. We first show that $v$ is trapped by two tangent planes at $0$.

Let $x_{1,k}$ and $x_{2,k}$ be the points on $\partial S_{h_{k},0}^{v}(0)$ where the hyperplanes perpendicular to the shortest axis of the John ellipsoid become tangent to $\partial S_{h_{k},0}^{v}(0)$, and let $p_{1,k}$ and $p_{2,k}$ denote subgradients at these points. Since

$$d_{1}(h_{k})d_{2}(h_{k})...d_{n}(h_{k})<Ch_{k}^{\frac{n+1}{2}},$$

we have that $d_{n}(h_{k})<\frac{C}{\delta^{n-1}}h_{k}$ for all $k$. By this observation and convexity we can rotate and pass to a subsequence such that

$$p_{1,k}\rightarrow c_{1}(\delta)e_{n},\quad p_{2}(k)\rightarrow-c_{2}(\delta)e_{n}.$$

Then $v$ is trapped by the planes $\pm c(\delta)x_{n}$. We conclude that

$$S_{h_{k},0}^{v}(0)\subset\{|x_{n}|<C(\delta)h_{k}\}.$$

To complete the proof, we show that the volumes of sections obtained with tilted supporting planes are too large. Take the largest $a$ such that $v\geq ax_{n}$ and consider the sections

$$S_{k}=S_{(1+aC(\delta))h_{k},a}^{v}(0).$$

Then $S_{k}$ engulf $S_{h_{k},0}^{v}(0)$. Furthermore,

$$\sup\{|x_{n}|:x\in S_{k}\}=R_{k}h_{k},$$

where $R_{k}\rightarrow\infty$ as $k\rightarrow\infty$. Indeed, if not, then for some small $\epsilon$ and a sequence $b_{i}\rightarrow 0$ we would have $v(x^{\prime},b_{i})>(a+\epsilon)b_{i}$ for all $x^{\prime}$. Convexity and $v(0)=0$ imply that $v>(a+\epsilon)x_{n}$ for all $x_{n}>b_{i}$, which in turn implies that

$$v>(a+\epsilon)x_{n},$$

contradicting the definition of $a$.

Finally, let $(x^{\prime}_{k},R_{k}h_{k})\in S_{k}$ be the point in $S_{k}$ furthest in the $e_{n}$ direction. Since $v$ grows at least quadratically, we have

$$|x^{\prime}_{k}|<C(\delta,a)h_{k}^{1/2}.$$

Recall that $S_{h_{k},0}^{v}(0)\subset\{|x_{n}|<C(\delta)h_{k}\}$. Since $d_{i}(h_{k})>\delta h_{k}^{1/2}$ for all $i\leq n-1$, $S_{k}$ contains the cone with vertex $(x^{\prime}_{k},R_{k}h_{k})$ and base given by a ball of radius $h_{k}^{1/2}(\delta-C(a,\delta)/R_{k})$ on the hyperplane $\{x_{n}=C(\delta)h_{k}\}$. We conclude that

$$|S_{k}|\geq c(\delta,a)R_{k}h_{k}^{\frac{n+1}{2}},$$

contradicting our definition of $\Sigma_{v}$ for $k$ large. ∎