Balan–Wang stability conjecture via ω(A)

Prove that there exist universal constants C > 0 and 0 < β < 1 such that for any M > 1 and any real matrix A ∈ R^{(2M−1)×M} in which every subset of M rows spans R^M, the quantity ω(A) satisfies ω(A) ≤ C (max_{k∈[N]} ||A_k||) β^M.

Background

The map x mod ±1 → |Ax| in real phase retrieval is injective if and only if ω(A) > 0 (the complement property). Stability is controlled by ω(A) and its decay with M. The conjecture posits an exponential decay bound in M, up to a row-norm factor, for matrices where any M rows span.

Such a bound would yield quantitative stability guarantees for minimal (2M−1) real phaseless measurements.

References

They also make the following intriguing conjecture. There exist universal constants $C>0$ and $0<\beta<1$ such that, for any $M>1$ and $A\in\mathbb{R}{2M-1\times M}$ a matrix for which any subset of $M$ rows spans $\mathbb{R}M$, \ \omega(A) \leq C \max_{k\in[N]}|A_k|\betaM, where $A_k$ is the $k$-th row of $A$.

— Randomstrasse101: Open Problems of 2025  (2603.29571 - Bandeira et al., 31 Mar 2026) in Conjecture, Section “Injectivity and Stability of Phase Retrieval (ASB)” (Entry 10)