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

We suggest a definition of the 1-parameter twisted affine Yangian $T{Y}_{\mathrm{\hslash }}(\widehat{\mathfrak{s}\mathfrak{o}}(n))$ and the 2-parameter twisted affine Yangian $T{Y}_{\mathrm{\hslash },\text{ve}}(\widehat{\mathfrak{s}\mathfrak{o}}(2n))$. These twisted affine Yangians are deformations of subalgebras of the universal central extension of $\mathfrak{s}\mathfrak{l}(2n)[u^{\pm 1},v]$. As for the 1-parameter twisted affine Yangian, we show that $T{Y}_{\mathrm{\hslash }}(\widehat{\mathfrak{s}\mathfrak{o}}(n))$ is a coideal of the affine Yangian associated with $\widehat{\mathfrak{s}\mathfrak{l}}(n)$. As for the two parameter twisted affine Yangian, we construct a homomorphism from $T{Y}_{\mathrm{\hslash },\text{ve}}(\widehat{\mathfrak{s}\mathfrak{o}}(8))$ to the universal enveloping algebra of the rectangular $W$-algebra ${\mathcal{W}}^k(\mathfrak{s}\mathfrak{p}(16),(2^8))$.