Conjectural standard homomorphism families from dihedral type to type D

Let $n$ be in the congruence classes specified below, let $K_1,K_2\subset[0,n]$, and let $w_0^S$ denote the longest element of the parabolic subgroup indexed by $S$. For $i\in\{1,2\}$, define the displayed images by deleting from $K_i$ the indicated arithmetic progression.

Type-D family conjecture. The following assignments define standard homomorphisms:

1. From $\operatorname{Br}^+(I_2(n+1))$ to $\operatorname{Br}^+(D_{n+1})$,

$$\widehat T_i\mapsto T_{w_0^{K_i\setminus(2i+4\mathbb Z_{\ge0})}},\qquad i\in\{1,2\},$$

for $(K_1,K_2)$ equal to $([1,n],\{0\}\cup[2,n-2])$ when $n\in1+4\mathbb Z_{>0}$, $([1,n],\{0\}\cup[2,n])$ when $n\in1+2\mathbb Z_{>0}$, or $([1,n-2],\{0\}\cup[2,n])$ when $n\in3+4\mathbb Z_{\ge0}$.

2. From $\operatorname{Br}^+(I_2(2n))$ to $\operatorname{Br}^+(D_{n+1})$,

$$\widehat T_i\mapsto T_{w_0^{K_i\setminus(2i-1+4\mathbb Z_{\ge0})}},\qquad i\in\{1,2\},$$

for $(K_1,K_2)$ equal to $([0,n],\{1\}\cup[4,n-2])$ when $n\in4\mathbb Z_{>0}$, $([0,n],\{1\}\cup[4,n])$ when $n\in2\mathbb Z_{>1}$, or $([0,n-2],\{1\}\cup[4,n])$ when $n\in2+4\mathbb Z_{>0}$.

The paper presents these as conjectural families of standard homomorphisms into type $D$, following the conjectural type-$A$ and type-$B$ families.