Consider the following result (which is Lemma 2.8 in Mitchell and Steel's paper on Fine Structure and Iteration Trees):

**Lemma 2.8** Let $\pi \colon \mathcal{H} \to \mathcal{M}$ be generalized $r \Sigma_{k}$
elementary, where $\mathcal M$ is a ppm (not of type III) and $1
\leq k < \omega$. Suppose that $\rho_{k}^{\mathcal{M}} \subseteq
\mathcal{H}$ and $\pi \restriction \rho_{k}^{\mathcal{M}} =
\operatorname{id}$. Suppose also that $\pi(r)$ is the $k$th standard parameter of
$(\mathcal{M}, \pi(q))$ and that $\pi(r)$ is $k$-solid and
$k$-universal over $(\mathcal{M}, \pi(q))$. Then

- $\rho_{k}^{\mathcal{H}} = \rho_{k}^{\mathcal{M}}$,
- $r$ is the $k$th standard parameter of $(\mathcal{H},q)$ and
- $r$ is $k$-universal over $(\mathcal{H}, q)$.

The proofs of items $1.$ and $2.$ are included in this paper and in both cases it seems that the $k$-solidity of $\pi(r)$ over $(\mathcal{M}, \pi(q))$ isn't actually needed. Hence I decided to prove item $3.$ to see how $k$-solidity comes into play. However, if my argument is correct, it doesn't rely on $k$-solidity either.

Here is my proof of item $3.$:

**Proof** *(of $3.$).* Let $A \in \mathcal{H}$ be such that $A \subseteq
\rho_{k}^{\mathcal{H}}$. Then $\pi(A) \cap \rho_{k}^{\mathcal{M}}
\in \mathcal M$. By the $k$-universality of $\pi(r)$ over
$(\mathcal{M}, \pi(q))$ there is hence some generalized Skolem term
$\tau \in S_{\kappa}$ and some $\vec{\alpha} \in ^{< \omega}
\rho_{k}^{\mathcal{M}}$ s.t.
$$
\pi(A) \cap \rho_{k}^{\mathcal{M}} =
\tau^{\mathcal{M}}[\vec{\alpha}, \pi(r), \pi(q)] \cap \rho_{k}^{\mathcal{M}}.
$$
Let $B := \tau^{\mathcal{H}}[\vec{\alpha},r,q] \cap \rho_{k}^{\mathcal{H}}$. Combining the fact
that $\pi$ is generalized $r \Sigma_{k}$-elementary, $\rho_{k}^{\mathcal{H}}=
\rho_{k}^{\mathcal{M}} \subseteq \mathcal{H}$ and $\pi \restriction
\rho_{k}^{\mathcal M} = \operatorname{id}$ we have that
\begin{align*}
\mathcal{H} \models A \cap \rho_{k}^{\mathcal{H}} = B &\iff \mathcal{H} \models A \cap \rho_{k}^{\mathcal{H}} = \tau^{\mathcal
H}[\vec{\alpha},
r,q] \cap
\rho_{k}^{\mathcal{H}}
\\
&\iff
\mathcal{M}
\models
\pi(A) \cap
\rho_{k}^{\mathcal{M}}
=
\tau^{\mathcal{M}}[\vec{\alpha},\pi(r),\pi(q)]
\cap \rho_{k}^{\mathcal{M}}.
\end{align*}
Since the last line is true, it follows that
$A \cap \rho_{k}^{\mathcal{H}}
=\tau^{\mathcal{H}}[\vec{\alpha},r,q] \cap \rho_{k}^{\mathcal{H}}$. Thus $r$ is indeed
$k$-universal over $(\mathcal{H}, q)$. *Q.E.D.*

**Question**: Did I miss something and this result actually relies on the $k$-solidity of $\pi(r)$ over $(\mathcal{M},\pi(q))$ or can this assumption be dropped?

If $k$-solidity is needed, I'd like to understand where exactly in the proof it is used and ideally I'd like to see an example in which Lemma 2.8 fails without $k$-solidity.

PS: I am aware that this question isn't exactly 'ongoing research' and I strongly considered posting it over at MSE. However, since the group of people able to answer this question is more likely to be encountered here and since a somewhat similar question has been asked and well-received here, I decided to go with mathoverflow.