Show that $p^T \Sigma^{-1} p \leq k^2$ is equivalent to $w^Tpp^Tw \leq k^2 w^T \Sigma w$

Question

I am trying to understand a research paper 1 (Eq 11 and Eq 10) which states the solution to the following 2 optimization problems are equivalent. $$\begin{array}{ll} & \min_{p} p^Tw \\ & \qquad \text{subject to } p^T \Sigma^{-1} p \leq k^2 \\ \equiv &\min_p p^Tw \\ & \qquad \text{subject to } p^Tw \leq k \sqrt{w^T\Sigma w} \end{array}$$ where $p, w \in R^{N\times 1}$ and $\Sigma \in R^{N \times N}$. $\Sigma$ is positive definite and $k \ge 0$. I tried solving the first problem using method of multipliers but couldn't show the equivalence. I finally get something like this \begin{array}{ll} k \Sigma w = p \sqrt{w^T\Sigma w} \end{array} and I am unable to show that this is equivalent to constraint in problem 2. Using basic trace properties, I was able to show that \begin{array}{ll} p^T \Sigma^{-1}p \leq K^2 N \end{array} but I need \begin{array}{ll} p^T \Sigma^{-1}p \leq K^2 \end{array} Can someone please suggest some ways of showing the equivalence?