Suggestion of a definition of the twisted affine Yangian of type $D$

Abstract: We suggest a definition of the twisted affine Yangian $TY_{\hbar,\varepsilon}(\widehat{\mathfrak{so}}(2n))$. We show that $TY_{\hbar,\varepsilon}(\widehat{\mathfrak{so}}(2n))$ coincides with the universal enveloping algebra of a subalgebra of the universal central extension of $\mathfrak{sl}(2n)[u^{\pm1},v]$ when we set two parameters $\hbar=\varepsilon=0$. We also construct a homomorphism from $TY_{\hbar,\varepsilon}(\widehat{\mathfrak{so}}(8))$ to the universal enveloping algebra of the rectangular $W$-algebra $\mathcal{W}^k(\mathfrak{sp}(16),(2^8))$.