Optimal reproducing-filter norm conjecture for shift-invariant subspaces

Let $s\in\mathbb{N}$ be odd and let $n\in\mathbb{N}$ satisfy $2n+1\geq s$. An $s$-dimensional shift-invariant subspace (SIS) is a shift-invariant subspace of the sequence space, and a filter $\varphi\in\mathbb{C}_n(\mathbb{Z})$ is reproducing for an SIS $X$ when $\varphi*x=x$ for every $x\in X$. Write $\mathcal{F}_n[\varphi]$ for the corresponding discrete Fourier transform.

Optimal reproducing-filter norm conjecture. Every $s$-dimensional SIS admits a reproducing filter $\varphi\in\mathbb{C}_n(\mathbb{Z})$ such that

$$\|\mathcal{F}_n[\varphi]\|_p\sqrt{2n+1}\leq s^{1/p}$$

for all $p\in[1,+\infty]$.

The conjecture asserts that the periodic examples described in the preceding remark are hardest possible, up to no loss in the constant, and would sharpen the preceding near-optimal construction for reproducing filters.