Isometric embeddings into $C(K)$-spaces doing stable phase retrieval

Motivated by a question posed by Freeman, Oikhberg, Pineau and Taylor, we prove that if $K$ is a compact Hausdorff space with $K^{(\alpha )}\mathrm{\ne }\mathrm{\varnothing }$, where $2<\alpha <\omega$, then $C[1,{\omega }^{\alpha }]$ isometrically embeds into $C(K)$ doing stable phase retrieval (SPR). We also show that the latter cannot be extended to the case $\alpha =2$.