Let $\Phi$ be the CDF of a standard Gaussian distribution, i.e. $$\Phi(x):=\int_{-\infty}^x \frac{1}{\sqrt{2\pi}} e^{-y^2/2}dy,\quad \forall~ x\in \mathbb R.$$ Denote by $\Phi^{-1}$ its inverse function. For every $p>0$, define $F_p: \mathbb R\times \mathbb R_+\to \mathbb R_+$ by

$$F_p(m,\sigma):=\left(\int_0^1 \big|(\sigma-1)\Phi^{-1}(t)+m\big|^p dt\right)^{1/p},\quad \forall~ m\in \mathbb R,~ \sigma\in \mathbb R_+.$$

Does there exist constants $c(p)\equiv c <C\equiv C(p)$ (only depending on $p$) such that

$$cF_2(m,\sigma) \le F_p(m,\sigma) \le CF_2(m,\sigma),\quad \forall~ m\in \mathbb R,~ \sigma\in \mathbb R_+?$$