solution to the problem $$\begin{cases}
  \Delta_b^2 U=0,&\text{in }{{\mathbb R}}^{N+1}_+={{\mathbb R}}^N\times(0,+\infty),\\
U(x,0)=f(x),&\text{in }{{\mathbb R}}^{N},\\
\lim_{t\to 0^+}t^{b}\frac{\partial U}{\partial
  t}=0 ,&\text{in }{{\mathbb R}}^{N}.
\end{cases}$$ Setting $V=\Delta_b U$ and taking into account , the above fourth order problem can be rewritten as the system $$\begin{cases}
  \Delta_b U=V,&\text{in }{{\mathbb R}}^{N+1}_+,\\
  \Delta_b V=0,&\text{in }{{\mathbb R}}^{N+1}_+,\\
U(x,0)=f(x),&\text{in }{{\mathbb R}}^{N},\\
\lim_{t\to 0^+}t^{b}\frac{\partial U}{\partial
  t}=0 ,&\text{in }{{\mathbb R}}^{N},\\
K_{s} \lim_{t\to 0^+}t^{b}\frac{\partial V}{\partial
  t}=(-\Delta)^s f ,&\text{in }{{\mathbb R}}^{N}.
\end{cases}$$ In [@Y] an Almgren’s frequency formula in the spirit of [@almgren] is derived for solutions to the higher order system $$\label{eq:51}
 \begin{cases}
  \Delta_b U=V,&\text{in }{{\mathbb R}}^{N+1}_+,\\
  \Delta_b V=0,&\text{in }{{\mathbb R}}^{N+1}_+,\\
\lim_{t\to 0^+}t^{b}\frac{\partial U}{\partial
  t}=0 ,&\text{in }{{\mathbb R}}^{N},\\
\lim_{t\to 0^+}t^{b}\frac{\partial V}{\partial
  t}=0 ,&\text{in }{{\mathbb R}}^{N},
\end{cases}$$ obtained by extending $s$-harmonic functions; in the spirit of Garofalo and Lin [@GL], such monotonicity formula allows proving a unique continuation property for solutions to system . In [@Y] a strong unique continuation property is also stated for $s$-harmonic functions.

The main goal of the present paper is to extend, in the case $s=\frac32$, the monotonicity formula developed in [@Y] for the homogeneous case to systems with a Neumann boundary coupling of the type $$\label{eq:52}
 \begin{cases}
  \Delta U=V,&\text{in }{{\mathbb R}}^{N+1}_+,\\
  \Delta V=0,&\text{in }{{\mathbb R}}^{N+1}_+,\\
\frac{\partial U}{\partial
  