Incomplete PIIDG orthogonal-transform conjecture

Let $\bm{U}$ be an orthogonal matrix and let $\bm{U}_1$ be an $N\times(N-m)$ block of $\bm{U}$. Let $\bm{b}_1\in\mathbb{R}^{N-m}$ be a PIIDG vector, meaning a partially independent and identically distributed Gaussian vector, and assume that $\bm{U}$ and $\bm{b}_1$ are mutually independent. Incomplete PIIDG orthogonal-transform conjecture. When $m/N\to0$, the following claims hold: (i) $\bm{U}_1\bm{b}_1$ converges to a PIIDG vector; (ii)

$$\operatorname{E}\{\|\bm{U}_1\bm{b}_1\|^2\}\to\operatorname{E}\{\|\bm{b}_1\|^2\};$$

and (iii) $\bm{U}_1\bm{b}_1$ is asymptotically independent of $\bm{U}_1$.

This extends the corresponding incomplete orthogonal-transform property from IIDG vectors to PIIDG vectors. The source supplies no evidence resolving the conjecture.