Demeter's book Fourier Restriction, Decoupling, and Applications give a principle that one cannot decouple in a direction where the manifold is flat. Which is the below proposition:

Proposition 9.5 Let $T$ be a tube in $\mathbb{R}^n$. Consider a partition $\mathcal{T}_N$ of $T$ into $N$ shorter, essentially congruent tubes. Then $${\rm{Dec}}(\mathcal{T}_N,p)\sim N^{1/2-1/p},$$ with implicit similarity constants independent of $N$.

The proof claim that considering the function $f$ whose fourier transform is a smooth approximation of $1_T$ can get the lower bound.

I have tried to assume that $T=\left\{(x_1,x^*)\in\mathbb{R}^n: x_1\in[0,1],x^*\in\mathbb{R}^n\right\}$. So $\mathcal{T}_N=\left\{T_1,T_2,\cdots,T_N\right\}$, where $T_k=\left\{(x_1,x^*)\in\mathbb{R}^n: x_1\in\left[\frac{k-1}{N},\frac{k}{N}\right],x^*\in\mathbb{R}^n\right\}$, $k=1,2,\cdots,N$. And $\left\{f_\varepsilon\right\}_{\varepsilon>0}\subset \mathcal{S}(\mathbb{R}^n)$ such that $1_T\leq \hat{f_\varepsilon}\leq 1_{(1+\varepsilon)T}$.

But for $k=2,3,\cdots,N-1$, $\mathcal{P}_{T_k}f_\varepsilon(x)=\int_{T_K}\hat{f_\varepsilon}(\xi) e^{2\pi i x\cdot\xi}{\operatorname{d}}\xi=\int_{\frac{k-1}{N}}^{\frac{k}{N}}e^{2\pi i x_1\xi_1}{\operatorname{d}}\xi_1\cdot\int_{\mathbb{R}^{n-1}}e^{2\pi i x^*\cdot\xi^*}{\operatorname{d}}\xi^*$ is disconverge.