The setup is as follows:

- $k/\mathbb{Q}_p$ is a finite extension, $\mathfrak{p}$ is the maximal ideal of $\mathcal{O}_k$, $q=\#(\mathcal{O}_k/\mathfrak{p})$
- $k'/k$ is a finite unramified extension of degree $d$

It's known that for a relative Lubin-Tate formal group $\mathcal{F}$ relative to $k'/k$ with parameter $\xi$ ($\xi\in\mathcal{O}_k$ with $\mathrm{ord}_{\mathfrak{p}}(\xi)=d$), it gives an abelian extension tower $k'(\mathcal{F}[\mathfrak{p}^n])$ of $k$ with degree $[k'(\mathcal{F}[\mathfrak{p}^n]):k']=(q-1)q^{n-1}$ and each $k'(\mathcal{F}[\mathfrak{p}^n])/k'$ is totally ramified.

My question is, if a tower $\{k_n'\}$ is given such that for any $n$, $k_n'/k$ is abelian, $k_n'/k'$ is totally ramified, and $[k_n':k']=(q-1)q^{n-1}$, then can we find a relative Lubin-Tate formal group $\mathcal{F}$ such that $k_n'=k'(\mathcal{F}[\mathfrak{p}^n])$? If not, are there any criteria for it?

In fact I'm considering the following special case: let $K$ be an imaginary quadratic field, $p$ be a prime split in $K$, $\mathfrak{p}$ be a prime of $K$ above $p$, $H$ be the Hilbert class field of $K$, $w$ be a prime of $H$ above $\mathfrak{p}$. Let $H_n$ be the ring class field of $K$ of conductor $p^n$, then $w$ is totally ramified over $H_n/H$ and we have $[H_n:H]=2(p-1)p^{n-1}/\#\mathcal{O}_K^\times$. I'd like to know the answer of above question for $k=K_{\mathfrak{p}}\cong\mathbb{Q}_p$, $k'=H_w$, $k_n'=H_{n,w}$, i.e. whether the anti-cyclotomic tower $\{H_{n,w}\}$ comes from (relative) Lubin-Tate formal group. (Of course it's true for cyclotomic tower and the $\mathbb{Z}_p$ extension tower unramified outside $\mathfrak{p}$. And it looks like that we are in trouble when $d_K=-3,-4$.)

EDIT: If we assume $k=\mathbb{Q}_p$ (as in the special case I'm considering) and $p\geq 3$, then by the fact $\mathbb{Q}_p^{\mathrm{ab}}=\mathbb{Q}_p^{\mathrm{unr}}(\mu_{p^\infty})$ and that the open subgroup of $\mathrm{Gal}(\mathbb{Q}_p^{\mathrm{unr}}(\mu_{p^\infty})/\mathbb{Q}_p^{\mathrm{unr}})\cong\mathbb{Z}_p^\times$ of index $(p-1)p^{n-1}$ is unique, we can conclude that $\mathbb{Q}_p^{\mathrm{unr}}k_n'=\mathbb{Q}_p^{\mathrm{unr}}(\mu_{p^n})$ for any $n$. Can we obtain more information from this?