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I believe that Daniele Tampieri's idea of using completeness Fichera's Theorem is the right one. However I think there is a simpler proof. In view of the Hahn-Banach Theorem, proving the density of the space \begin{equation} S= \left\{(f,f|_{\partial D})\ |\ f\in C^{\infty}(\overline{D}), \Delta f|_{\partial D} =0\right\} \end{equation} in $L^{2}(D)\times L^{2}(\partial D)$ is equivalent to show that, if $(F,G)\in L^{2}(D)\times L^{2}(\partial D)$ is such that \begin{equation} \int_{D}F\, f\, dx + \int_{\partial D}G\, f\, d\sigma = 0 \end{equation} for any $f\in S$, then $F=0$ a.e. in $D$ and $G=0$ a.e. on $\partial D$. So suppose that these orthogonality conditions hold. In particular we have $$ \int_{D}F\, f\, dx = 0 $$ for any $f\in C^{\infty}_{c}(D)$. It is well known that this implies $F=0$ a.e. in $D$. Therefore $$ \int_{\partial D}G\, f\, d\sigma = 0 $$ for any $f\in S$. Harmonic polynomials obviously belong to $S$ and then $$ \int_{\partial D}G\, \omega\, d\sigma = 0 $$ for any harmonic polynomial $\omega$. Fichera's theorem implies $G=0$ a.e. on $\partial D$.