 right-hand side is non-negative and, hence, the left-hand side is an upper bound for each term on the right-hand side.
It also implies that $B$ is positive and then
\begin{equation*}
   
E_1(B)\equiv E_{K+1}(R^\Omega_\alpha)+ \alpha^2 +\dfrac{(M+a)\log\alpha}{\alpha}\ge 0.
\end{equation*}
By \eqref{eq-ap12}, for any fixed $n\in\mathbb N$ there is $\mu_n>0$ independent of $\alpha$ such that
\begin{equation}
\label{hyp1}
0\le E_n(B)\le \mu_n, \quad \big( 1+ E_n(B)\big)^{-1} \ge (1+\mu_n)^{-1}.
\end{equation}

In order to construct a suitable identification map $J:\mathop{\mathcal{Q}}(B)\to \mathop{\mathcal{Q}}(B')$ we pick functions
$\rho^\pm_j\in C^1\big([0,\ell_j]\big)$ such that
\[
\rho^+_j=\begin{cases}
1 & \text{ in a neighborhood of $0$},\\
0 & \text{ in a neighborhood of $\ell_j$},
\end{cases}
\quad
\rho^-_j=\begin{cases}
0 & \text{ in a neighborhood of $0$},\\
1 & \text{ in a neighborhood of $\ell_j$},
\end{cases}
\quad j\in \mathcal{J}_*,
\]
and then choose a constant $\rho_0>0$ such that
\begin{equation}
   \label{eq-rho0}
\|\rho^\pm_j\|_{L^\infty(0,\ell_j)}+\|(\rho^\pm_j)'\|_{L^\infty(0,\ell_j)}\le \rho_0 \text{ for all }j\in \mathcal{J}_*.
\end{equation}
Recall that 
$I_{j,\delta}:=(\lambda_j\delta,\ell_j-\lambda_{j+1}\delta)=:(\iota_j,\tau_j)$, hence,
\[
{\rho_j^+(\iota_j)=1, \quad \rho_j^+(\tau_j)=0, \quad
\rho_j^-(\iota_j)=0, \quad \rho_j^-(\tau_j)=1},
\]
as $\alpha$ is sufficiently large. Therefore, the map
\begin{gather*}
J:\mathop{\mathcal{Q}}(B)\to \mathop{\mathcal{Q}}(B')\equiv \mathop{\oplus}\nolimits_{j=1}^M H^1_0(I_{j,\delta}),
\quad J u= (J_j u),\\
(J_j u)(s):=(P_j u)(s)-(P_j u)(\iota_j)\rho^+_j(s) - (P_j u)(\tau_j)\rho^-_j(s)
\end{gather*}
is well-defined. We estimate, using the Cauchy-Schwarz inequality,
\begin{equation}
   \label{pj002}
\begin{aligned}	
\big|P_j u(\iota_j)\big|^2+\big|P_j u(\tau_j)\big|^2
&=\Big( \int_0^\delta \psi(t)\,w_j\big(\Phi_j(\iota_j,t)\big)\,\mathrm{d} t\Big)^2
+\Big( \int_0^\delta \psi(t) \,w_j\big(\Phi_j(\tau_j,t)\big)\,\mathrm{d} t\Big)^2\\
&\le \int_0^\delta w_j\big(\Phi_j(\iota_j,t)\big)^2\, dt + \int_0^\delta w_j\big(\Phi_j(\tau_j,t)\big)^2\,\mathrm{d} t
\equiv \int_{\partial_\mathrm{ext} W_{j,\delta}} w_j^2\,\mathrm{d} s.
\end{aligned}
\end{equation}
Recall the inequality $(x+y)^2\ge(1-\varepsilon)x^2-y^2/\varepsilon$ valid for any $x,y\in\mathbb{R}$ and $\varepsilon>0$. Then
\begin{align*}
\|J_j u\|^2&=\int_{I_{j,\delta}} \Big|(P_j u)(s)-(P_j u)(\iota_j)\rho^+_j(s) - (P_j u)(\tau_j)\rho^-_j(s)\Big|^2\,\mathrm{d} s\\
&\ge (1-\varepsilon)\int_{I_{j,\delta}} \Big|(P_j u)(s)\Big|^2\,\mathrm{d} s-\dfrac{1}{\varepsilon}\int_{I_{j,\delta}}\Big|(P_j u)(\iota_j)\rho^+_j(s) + (P_j u)(\tau_j)\rho^-_j(s)\Big|^2\,\mathrm{d} s,
\end{align*}
and, using \eqref{pj002} and the constant $\rho_0$ from \eqref{eq-rho0} we have
\begin{align*}
&\begin{aligned}
\int_{I_{j,\delta}}\Big|(P_j u)(\iota_j)\rho^+_j(s) & + (P_j u)(\tau_j)\rho^-_j(s)\Big|^2\,\mathrm{d} s\\
&\le 2\ell_j\rho_0^2 \Big( \big|P_j u(\iota_j)\big|^2+\big|P_j u(\tau_j)\big|^2 \Big)\le 2 \ell_j \rho_0^2 \int_{\partial_\mathrm{ext} W_{j,\delta}} w_j^2\,\mathrm{d} s,
\end{aligned}\\
&\|J_j u\|^2\ge (1-\varepsilon)\|P_j u\|^2-\dfrac{2 \ell \rho_0^2}{\varepsilon}\int_{\partial_\mathrm{ext} W_{j,\delta}} w_j^2\,\mathrm{d} s,
\quad \ell:=\max \ell_j.
\end{align*}
Therefore, using \eqref{eq-uvw}  we arrive at
\begin{multline*}
\|u\|^2-\|Ju\|^2=\sum_{j=1}^M \|v_j\|^2+\sum_{j=1}^M \|w_j\|^2+\|u_c\|^2 -\sum_{j=1}^M \|J_j u\|^2\\
\begin{aligned}
&\le \sum_{j=1}^M \|v_j\|^2+\sum_{j=1}^M \|w_j\|^2+\|u_c\|^2-(1-\varepsilon)\sum_{j=1}^M\|P_j u\|^2 
+\dfrac{2 \ell \rho_0^2}{\varepsilon}\sum_{j=1}^M\int_{\partial_\mathrm{ext} W_{j,\delta}} w_j^2\,\mathrm{d} s\\
&=\sum_{j=1}^M \|v_j\|^2+\sum_{j=1}^M\big( \|w_j\|^2- \|P_j u\|^2 \big)+\varepsilon\sum_{j=1}^M\|P_j u\|^2
+ \dfrac{2 \ell \rho_0^2}{\varepsilon}\sum_{j=1}^M\int_{\partial_\mathrm{ext} V_{j,\delta}} v_j^2\,\mathrm{d} s+\|u_c\|^2\\
&\le \sum_{j=1}^M \|v_j\|^2+\sum_{j=1}^M\big( \|w_j\|^2- \|P_j u\|^2 \big)+ \varepsilon \|u\|^2 + \dfrac{2 \ell \rho_0^2}{\varepsilon}\sum_{j=1}^M\int_{\partial_\mathrm{ext} V_{j,\delta}} v_j^2\,\mathrm{d} s+\|u_c\|^2.
\end{aligned}
\end{multline*}
Using \eqref{eq-buu2} we obtain an upper bound for all terms on the right-hand side except $\varepsilon \|u\|^2$:
\[
\|u\|^2-\|Ju\|^2\le \Big(\dfrac{2a\log^2\alpha}{\alpha^2}+{\dfrac{2}{\alpha^2}}+\dfrac{4 \ell \rho_0^2 a\log^2\alpha}{\varepsilon \alpha}\Big) B[u,u]+\varepsilon \|u\|^2.
\]
Taking $\varepsilon:=\log\alpha/\sqrt{\alpha}$ and choosing $c_1>0$ sufficiently large we obtain
\begin