Conjectural type-A standard homomorphisms with a generator times the longest element

Fix $n$ and $i\in[2,n-2]$. Let $T_i$ be the standard generator and let $w_0^{[1,n-1]}$ be the longest element of the parabolic subgroup on $[1,n-1]$ in type $A_n$.

Type-A twelve-period homomorphism conjecture. Define

$$\widehat{T}_1\mapsto T_iT_n,\qquad \widehat{T}_2\mapsto T_{w_0^{[1,n-1]}}.$$

Then these assignments define a standard homomorphism

$$\operatorname{Br}^+(I_2(12/d))\to\operatorname{Br}^+_{n+1},$$

where

$$d=1+\delta_{2i,n}+\delta_{2i,n-1}+\delta_{2i,n+1}.$$

This is listed as another conjectural family of non-disjoint standard homomorphisms; the paper states that the families in this section were verified for $n\le15$.