The goal of this paper is to address the following question: if
A
A
is an
E
k
\mathbf {E}_{k}
-ring for some
k
≥
1
k\geq 1
and
f
:
π
0
A
→
B
f\colon \pi _0 A \to B
is a map of commutative rings, when can we find an
E
k
\mathbf {E}_{k}
-ring
R
R
with an
E
k
\mathbf {E}_{k}
-ring map
g
:
A
→
R
g\colon A \to R
such that
π
0
g
=
f
\pi _0 g = f
? A classical result in the theory of realizing
E
∞
\mathbf {E}_\infty
-rings, due to Goerss–Hopkins, gives an affirmative answer to this question if
f
f
is étale. The goal of this paper is to provide answers to this question when
f
f
is ramified. We prove a non-realizability result in the
K
(
n
)
K(n)
-local setting for every
n
≥
1
n\geq 1
for
H
∞
H_\infty
-rings containing primitive
p
p
th roots of unity. As an application, we give a proof of the folk result that the Lubin–Tate tower from arithmetic geometry does not lift to a tower of
H
∞
H_\infty
-rings over Morava
E
E
-theory.