Uniqueness conjecture for rank-one simple transitive 2-representations

Let $D$ be the algebra of dual numbers, let $\mathscr{D}$ be the associated 2-category of projective bimodules, and for each positive integer $k$ let $\mathcal{J}_k$ denote the two-sided cell of string bimodules with $k$ valleys. A simple transitive 2-representation has rank $1$ when its underlying category has one isomorphism class of indecomposable objects, and has apex $\mathcal{J}_k$ when $\mathcal{J}_k$ is its maximal two-sided cell acting nontrivially. Uniqueness conjecture. For each $k\geq 1$, there exists a unique, up to equivalence, simple transitive 2-representation of $\mathscr{D}$ of rank $1$ with apex $\mathcal{J}_k$. The theorem preceding this conjecture establishes the rank-one case for $k=1$ only; uniqueness for all $k\geq 1$ is the remaining assertion.