Infinite not contact isotopic embeddings in $(S^{2n-1},\xi_{\mathrm{std}})$ for $n\ge 4$

For $n\ge 4$, we show that there are infinitely many formally contact isotopic embeddings of $(ST^*S^{n-1},\xi_{\mathrm{std}})$ to $(S^{2n-1},\xi_{\mathrm{std}})$ 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.