Periodicity conjecture for symmetric-power quotients of $Q^*$

Let $d,e,n\in\mathbb{N}$ satisfy the hypotheses of Corollary $$. Write $Q^m[r]$ for the indicated filtration term, and let

$$\bullet 1^{e-d}:G_0\left(\mathscr{F}_n^{\geq d-(n-d)}\right)\stackrel{\cong}{\longrightarrow}G_0\left(\mathscr{F}_{n+e-d}^{\geq e-(n-d)}\right)$$

be the isomorphism of Grothendieck groups.

Periodicity conjecture. Under this isomorphism,

$$\left[Q^n/Q^n[d-1]\right]\longmapsto\left[Q^{n+e-d}/Q^{n+e-d}[e-1]\right].$$

The conjecture predicts periodicity of these symmetric-power quotients under representation stability. It is known when $n-d\leq 5$, while the general case remains open; the stated hypotheses include the stability conditions from Corollary $$.