The conjecture identifying diagrammatic and exceptional-point irreducible 2BTL representations

Let $N$ be the system size, let $n$ be an allowed label, and let $\epsilon_1,\epsilon_2\in\{+1,-1\}$. Let $W^{(N,n)}_{\epsilon_1,\epsilon_2}$ denote the irreducible representation of the two-boundary Temperley–Lieb algebra obtained from the diagrammatic approach, and let $V^{(N,n)}_{\epsilon_1,\epsilon_2}$ and $V^{(N)}_{\epsilon}$ denote the irreducible representations obtained from the $2^N$-dimensional representation at exceptional parameter values. Set

$$y=-n+\epsilon_1x_1+\epsilon_2x_2.$$

2BTL representation-identification conjecture. For $n>0$, the irreducible representation $W^{(N,n)}_{\epsilon_1,\epsilon_2}$ is equivalent to $V^{(N,n)}_{\epsilon_1,\epsilon_2}$, and

$$W^{(N,0)}_{+,+}\simeq V^{(N)}_{+}.$$

The conjecture is motivated by the coincidence between the dimensions of the diagrammatic irreducible representations and those obtained from the exceptional points. The source gives no resolution, so the identification remains open here.