Monotonicity conjecture for average singular values of Gaussian matrices

Bandeira, Kennedy, and Singer conjectured that the normalized average singular value decreases with dimension for complex Gaussian matrices but increases for real Gaussian matrices. The complex proof history includes a 2016 attempt that was later withdrawn; newer papers claim proofs of both halves.

Known results

• The complex sequence was known to converge to $\frac{8}{3\pi}$, while the real monotonicity remained open in the earlier literature.
• Abreu’s 2016 complex proof attempt was revised through 2023 and then marked withdrawn.
• A later paper by Hutník claims $\alpha_{\mathbb{C}}(N+1)<\alpha_{\mathbb{C}}(N)$ for every $N\geq 1$, with extensions to rectangular complex matrices.
• The complex decrement is reported to satisfy $\Delta_d\sim \frac{\log d}{8\pi d^3}$.

2026 claims for both directions

A 2026 preprint claims $\alpha_{\mathbb{R}}(N+1)-\alpha_{\mathbb{R}}(N)>\frac{1}{1000N^2}$ for every $N\geq 1$, which would prove strict real increase. Together with the newer complex result, this would settle the conjecture; the claims remain unverified.

Current status (as of September 2026): Both inequalities have proof claims, but the complex claim follows a withdrawn earlier proof attempt and neither direction has independent verification.