Conjectural interval-family homomorphisms from dihedral type to type A

Let $1\le b\le a<n-1$ and $a>1$. For the type-$A$ braid monoid, write $w_0^{[r,s]}$ for the longest element of the parabolic subgroup on the interval $[r,s]$.

Interval-family homomorphism conjecture. The assignments

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

define a homomorphism $\operatorname{Br}^+(I_2(N))\to\operatorname{Br}^+_n$ if one of the following holds:

$$N=\frac{2n}{n+b-a-2}\in2\mathbb Z_{>0};$$ $$a+b=n\quad\text{and}\quad N=\frac{n}{b-1}\in1+2\mathbb Z_{>0};$$ $$a+1=2(b-1),\quad a+b<n,\quad\text{and}\quad N=6.$$

The paper presents these as a conjectural infinite family of non-disjoint standard homomorphisms and reports verification for $n\le15$.