\begin{corollary}[shallow pockets off the tip]\label{cor:shallow}
Let $(W_n,B_n)$ be Blaschke models as in Theorem~\ref{thm:C}, after its rotations, and let $I$ be a closed arc
of the circle that avoids $\theta=\pi$ \textup{(}possibly a single point\textup{)}. Then
\[
  m_n:=\min_j\,\min_{e^{it}\in I}\bigl(\sigma_n-P_{a_j}\bigr)(e^{it})\ \longrightarrow\ \infty ,
\]
the first minimum being over all zeros $a_j$ of $B_n$. Consequently every off pocket of $(W_n,B_n)$ in $I$ has depth
$x<1/\min_I\sigma_n\le1/m_n$, every owned pocket in $I$ has depth $x\le1/m_n$, and at all these pockets
$|W_n'|\ge r_{\rm in}m_n\ge m_n/4$.
\end{corollary}

\begin{proof}
Let $\delta=\min(\operatorname{dist}(I,\pi),\pi/2)$, $c=\cos(\delta/2)$ and $\rho\in[1-2(\sqrt c-c),1)$. As in the proof of
Corollary~\ref{cor:offtip}, $\log4-2\log|1-\rho e^{it}|\ge-\log c>0$ for $e^{it}\in I$. With $X$ as in Lemma~\ref{lem:H}, the maximum of
$X(a)$ over the zeros of $B_n$ tends to $0$ uniformly in $t$, since the zeros leave every compact set. Fix $j$ and apply
Lemma~\ref{lem:H} to $B_n/B_{a_j}$, whose zeros are those of $B_n$ other than $a_j$; with $w=\rho e^{it}$,
\[
  \log\frac1{|B_n(w)|}-g_{\mathbb D}(w,a_j)=\log\frac1{|(B_n/B_{a_j})(w)|}\le4(1-\rho^2)\,(\sigma_n-P_{a_j})(e^{it}),
\]
and $g_{\mathbb D}(w,a_j)\le X(a_j)$ (proof of Lemma~\ref{lem:H}). Since $\log(1/|B_n(\rho e^{it})|)\to\log4-2\log|1-\rho e^{it}|$
uniformly in $t$, and neither error term depends on $j$ or on $t$, we get $\liminf_nm_n\ge-\log c/(4(1-\rho^2))$; let $\rho\to1$.
An off pocket has $\sigma_nx<1$, and an owned pocket has $(\sigma_n-P_{a_j})x\le1$ by Theorem~\ref{thm:own}(ii). At a pocket
$|v_n|=x$, so $|W_n'|=|W_n|/x$, and $|W_n|\ge r_{\rm in}\ge1/4$ on the circle.
\end{proof}

So on each fixed closed arc avoiding the tip, the owned pockets become shallow and sit where the harmonic density
$d\omega/ds=1/(2\pi|W_n'|)$ tends to $0$. If the owned pockets accumulate at the tip, the corollary says nothing about them.