Let $G_{(n)}=(G_1,\ldots,G_n)$ be a vector of $n$ i.i.d. normal vars. How to show $(d+1)(d-1)E[1/||G_{(d+1)}||_1^2]\ge d(d-2)E[1/||G_{(d)}||_1^2]?$

Question

Let $G_{(n)}=(G_1,G_2,\ldots,G_{n})$ be a vector of $n$ normal i.i.d. random varibles ($G_i\sim\mathcal N(0,1)$).

How can we show that for all $d\in\mathbb N^+$: $$ (d+1)\cdot (d-1)\cdot \mathbb E[1/||G_{(d+1)}||_1^2] \ge d\cdot (d-2)\cdot \mathbb E[1/||G_{(d)}||_1^2]? $$

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This question is related to my previous question, where the above inequality would imply that $\mathbb E\left[\frac{{d+1}}{||a||_1^2}\right] \ge \mathbb E\left[\frac{ d}{||b||_1^2}\right]$, where $a\in S^d$ and $b\in S^{d-1}$ be points chosen uniformly at random on the $(d+1)$ and $d$-dimensional spheres.