Infinite not contact isotopic embeddings in $(S^{2n-1},{\xi }_{\mathrm{s}\mathrm{t}\mathrm{d}})$ for $n\ge 4$

For $n\ge 4$, we show that there are infinitely many formally contact isotopic embeddings of $(S{T}^{\ast }{S}^{n-1},{\xi }_{\mathrm{s}\mathrm{t}\mathrm{d}})$ to $(S^{2n-1},{\xi }_{\mathrm{s}\mathrm{t}\mathrm{d}})$ that are not contact isotopic. This resolves a conjecture of Casals and Etnyre except for the $n=3$ case. The argument does not appeal to the surgery formula of critical handle attachment for Floer theory/SFT.