This question is a generalization of the question Volume ratio of $\ell_1$ balls and $\ell_1$ surfaces

For any $p\in[1,\infty]$ define $\|x\|_p := (|x_1|^p+\cdots+|x_d|^p)^{1/p}$ for $p\in[1,\infty)$ and $\|x\|_\infty := \max_{1\leq i\leq d}|x_i|$ for $p=\infty$. Denote $B_p^d := \{x\in\mathbb R^d: \|x\|_p\leq 1\}$ as the unit $\ell_p$-ball in $d$ dimension, and let $\partial B_p^d := \{x\in\mathbb R^d: \|x\|_p = 1\}$. The quantity of interest is the following "ratio" $$ \mathfrak d_{p,d} := \frac{\mathrm{vol}_{d-1}(\partial B_p^d)}{\mathrm{vol}_{d}(B_p^d)}. $$

My question is the following:

For any fixed $p\in[1,\infty]$, how does $\mathfrak d_{p,d}$ asymptotically scale with dimension $d$, as $d\to\infty$?

For some special cases, both $\mathrm{vol}_d(B_p^d)$ and $\mathrm{vol}_{d-1}(\partial B_p^d)$ have closed form solutions and the ratio $\mathfrak d_{p,d}$ can be calculated explicitly. Below are three examples:

$p=2$: in this case $\mathrm{vol}_d(B_2^d) = \frac{\pi^{d/2}}{\Gamma(d/2+1)}$, $\mathrm{vol}_{d-1}(\partial B_2^d) = \frac{2\pi^{d/2}}{\Gamma(d/2)}$ and therefore $\mathfrak d_{2,d} = d$;

$p=1$: in this case $\mathrm{vol}_d(B_1^d) = \frac{2^d}{d!}$ and $\mathrm{vol}_{d-1}(\partial B_1^d) = \frac{2^d \sqrt{d}}{(d-1)!}$, and therefore $\mathfrak d_{1,d} = d\sqrt{d}$

$p=\infty$: in this case $\mathrm{vol}_d(B_\infty^d) = 2^d$ and $\mathrm{vol}_{d-1}(\partial B_\infty^d) = 2^d n$. Therefore $\mathfrak d_{\infty,d} = d$.

From the above examples, my (very wild) guess is that $\mathfrak d_{p,d} \asymp d^{1+1/p-1/2}$ for $1\leq p\leq 2$ and $\mathfrak d_{p,d} \asymp d$ for $p\geq 2$. But of course I could be very wrong. The challenge for general $p$ is the apparent difficulty in evaluating the volume of $L_p$ sphere areas (e.g., Surface area of an $\ell_p$ unit ball?), but I'm hoping that the (asymptotic) ratio $\mathfrak d_{p,d}$ for general $p$ is potentially easier to evaluate.

**Edit:** I realized that, because the volume of a unit $\ell_p$ *ball* does have closed forms for general $p\in[1,\infty]$, the question can be answered if we know the **asymptotic dependency of dimension $d$ of the (unit) $\ell_p$-surface area**. This question was explicitly mentioned in the comment under the accepted answer of the following question
Surface area of superellipsoid (dice)
Unfortunately, the OP in that question did not pursue this direction.