PIIDG extension of the orthogonality lemma for separable-IID functions

Let $\bm{a}$ and $\bm{u}$ be PIIDG vectors, meaning partially independent and identically distributed Gaussian vectors, and assume that they are entry-wise jointly Gaussian. Let $\bm{\boldsymbol\pi}=\pi(\bm{a})$ be generated by a separable-IID function, and suppose

$$\operatorname{E}\{\bm{a}\}=\operatorname{E}\{\bm{u}\}=\bm{0},\qquad \operatorname{E}\{\bm{a}^{\rm T}\bm{\boldsymbol\pi}\}=0.$$

PIIDG orthogonality conjecture. Then

$$\frac{1}{N}\bm{a}^{\rm T}\bm{\boldsymbol\pi}\overset{\rm LLN}{\longrightarrow}0$$

and

$$\frac{1}{N}\bm{u}^{\rm T}\bm{\boldsymbol\pi}\overset{\rm LLN}{\longrightarrow}0.$$

This conjecture extends the stated extended Stein lemma from IIDG vectors to PIIDG vectors. The source gives no resolution or proof.