The domination conjecture for perturbative PSU(n)-invariants

Let $M$ be a rational homology 3-sphere, let $\tau^{PSU(n)}(M)$ denote its perturbative $PSU(n)$-invariant, and let $\tau_r^{SU(n)}(M)$ denote the quantum invariant at a positive integer level $r$.

Domination conjecture. The perturbative invariant $\tau^{PSU(n)}(M)$ dominates the quantum invariants $\tau_r^{SU(n)}(M)$ for every positive integer $r$, not necessarily prime.

The preceding theorem recovers the quantum invariant through the perturbative invariant after applying the reduction map for sufficiently large primes. This conjecture asks for the corresponding domination for all positive integer levels, including nonprime levels; the source gives no resolution.