Let $K\subset\mathbb{R}^d$ be a compact set with non-empty interior and Lipschitz boundary. In Section VI.3 of his book "Singular Integrals and Differentiability Properties of Functions", E. M. Stein constructs a linear operator $E_K$ continuously mapping the Sobolev space $W_{p,k}(K)$ into the Sobolev space $W_{p,k}(\mathbb{R}^d)$ for all $1\leq p\leq\infty$, $k=0,1,2,\ldots$, and such that $(E_K f)(x)=f(x)$ for all $x\in K$, $f\in W_{p,k}(K)$, nowadays called *Stein's extension operator* in the literature (actually, Stein builds $E_K$ with the above properties for $K$ closed with non-empty interior and locally Lipschitz boundary, but this will not be needed for the question). Consider now the linear operator $\tilde{E}_K:C^\infty(\mathbb{R}^n)\rightarrow C^\infty(\mathbb{R}^n)$ given by

$$\tilde{E}_K f=E_K(f|_K)$$

By Sobolev's embedding theorem (which does hold for $K$ as above), $\tilde{E}_K$ is a continuous linear map.

Question:Does the distribution kernel of $\tilde{E}_K$ have its wave front set contained in the conormal bundle to the diagonal of $\mathbb{R}^n\times\mathbb{R}^n$? In other words, does the extension of $\tilde{E}_K$ to $\mathscr{D}'(\mathbb{R}^n)$ by duality decrease wave front sets?