The three-family conjecture for standard homomorphisms from $I_2(N)$ to $A_n$

Let $1\le b\le a<n-1$ and $a>1$. Let $w_\circ^{[r,s]}$ denote the longest element associated with the interval $[r,s]$. Three-family conjecture for $I_2(N)$ homomorphisms. The assignments

$$\widehat T_1\mapsto T_{w_\circ^{[1,a]}},\qquad \widehat T_2\mapsto T_{w_\circ^{[b,n-1]}}$$

define a homomorphism $\operatorname{Br}^+(I_2(N))\to\operatorname{Br}^+_n$ if one of the following holds: $N=2n/(n+b-a-2)\in2\mathbb Z_{>0}$; $a+b=n$ and $N=n/(b-1)\in1+2\mathbb Z_{>0}$; or $a+1=2(b-1)$, $a+b<n$, and $N=6$. These are among the conjectural families introduced in the paper, which reports verification for $n\le15$; their general status remains open.