The conjectural families of standard homomorphisms to $D_{n+1}$

Use the displayed labeling of the Coxeter graph of type $D_{n+1}$, let $K_1,K_2\subset[0,n]$, and let $w_\circ^K$ denote the longest element associated with $K$. Conjectural $D_{n+1}$ families. The following assignments define standard homomorphisms. First, $\widehat T_i\mapsto T_{w_\circ^{K_i\setminus(2i+4\mathbb Z_{\ge0})}}$ for $i\in\{1,2\}$ defines a standard homomorphism $\operatorname{Br}^+(I_2(n+1))\to\operatorname{Br}^+(D_{n+1})$ for the three listed pairs: $K_1=[1,n]$, $K_2=\{0\}\cup[2,n-2]$ if $n\in1+4\mathbb Z_{>0}$; $K_1=[1,n]$, $K_2=\{0\}\cup[2,n]$ if $n\in1+2\mathbb Z_{>0}$; or $K_1=[1,n-2]$, $K_2=\{0\}\cup[2,n]$ if $n\in3+4\mathbb Z_{\ge0}$. Second, $\widehat T_i\mapsto T_{w_\circ^{K_i\setminus(2i-1+4\mathbb Z_{\ge0})}}$ defines a standard homomorphism $\operatorname{Br}^+(I_2(2n))\to\operatorname{Br}^+(D_{n+1})$ for the three listed pairs: $K_1=[0,n]$, $K_2=\{1\}\cup[4,n-2]$ if $n\in4\mathbb Z_{>0}$; $K_1=[0,n]$, $K_2=\{1\}\cup[4,n]$ if $n\in2\mathbb Z_{>1}$; or $K_1=[0,n-2]$, $K_1=\{1\}\cup[4,n]$ if $n\in2+4\mathbb Z_{>0}$. The paper presents these as conjectural families and reports verification for $n\le15$; no general proof or disproof is supplied here.