The $12/d$ family conjecture for standard homomorphisms to $A_{n+1}$

For $i\in[2,n-2]$, let $d=1+\delta_{2i,n}+\delta_{2i,n-1}+\delta_{2i,n+1}$. $12/d$ family conjecture. The assignments

$$\widehat T_1\mapsto T_iT_n,\qquad \widehat T_2\mapsto T_{w_0^{[1,n-1]}}$$

define a standard homomorphism

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

This is one of the conjectural infinite families in the paper, which reports verification for $n\le15$; no general proof or disproof is supplied here.