Uniqueness of the splitting at d=3

Prove uniqueness of the decomposition of polynomial mathfrak{sp}(2)-singlets into an AdS-traceless representative and an element of the mathfrak{sp}(2)-ideal for d=3.

Background

For d>3, the paper proves that every polynomial in the off-shell higher-spin algebra has a unique AdS-traceless representative modulo the mathfrak{sp}(2)-ideal. The uniqueness proof relies on a quasiprojector whose relevant family degenerates at d=3, where the AdS-traceless modules also split into self-dual and anti-self-dual sectors.

The paper establishes existence of the splitting at d=3 but does not establish uniqueness. Resolving this issue is necessary to obtain the same representative-based realization of the factor algebra in three dimensions.