The Young-diagram decomposition conjecture for tensor powers of the orthogonal adjoint representation

Let $\mathfrak{so}(N)$ be the orthogonal Lie algebra, and let $\operatorname{ad}$ denote its adjoint representation. For a Young diagram $\lambda$, write $\lambda\vdash 2m$ when it has $2m$ boxes, and let $[k^2]$ denote the rectangular diagram with two rows of length $k$. For sufficiently large $N$, consider the irreducible representations occurring in $\operatorname{ad}^{\otimes k}$. The Young-diagram decomposition conjecture. Every such irreducible representation is associated with a Young diagram $\lambda\vdash 2m$ with $m\leq k$ and with at most $k$ columns. All these diagrams are obtained from $[k^2]$ by allowable shifts of boxes to the left and allowable cancellations of pairs of boxes in one row. Here, an allowable cancellation is one that produces a Young diagram with fewer boxes. This conjecture proposes an explicit description of the irreducible representations in tensor powers of the orthogonal adjoint representation; its resolution status is not specified in the source.