Generic norm growth of powers of homogeneous unimodular Fourier multipliers

For an integer $d\ge 2$, $t\in \mathbb{R}$ and a $0$-homogeneous function $\mathrm{\Phi }\in C^{\mathrm{\infty }}({\mathbb{R}}^d\setminus \{0\},\mathbb{R})$, we consider the family of Fourier multiplier operators $T_{\mathrm{\Phi }}^t$ associated with symbols $\xi \mapsto \exp (it\mathrm{\Phi }(\xi ))$ and prove that for a generic phase function $\mathrm{\Phi }$, one has the estimate $\parallel T_{\mathrm{\Phi }}^t{\parallel }_{L^p\to L^p}{\gtrsim }_{d,p,\mathrm{\Phi }}⟨t{⟩}^{d\mid \frac{1}{p}-\frac{1}{2}\mid }$. That is the maximal possible order of growth in $t\to \pm \mathrm{\infty }$, according to the previous work by V. Kovač and the author and the result shows that the two special examples of functions $\mathrm{\Phi }$ that induce the maximal growth, given by V. Kovač and the author and independently by D. Stolyarov, to disprove a conjecture of Maz’ya actually exhibit the same general phenomenon.