Nearly valid higher-degree pseudomoment extension of the Montanari–Sen witness

Let $\delta \in (0,1)$, and let $\bm{M}^{(\delta)}$ be the Montanari–Sen witness. Fix $d \in 2\mathbb{N}$, and choose constants $\sigma_k^2$ depending only on $\delta$. Let $\bm{Z}^{\mathsf{mult}[k,k]}$ be the tensor construction defined in the surrounding discussion. Pseudomoment-extension conjecture. With high probability as $N\to\infty$, the entries of $\bm{Z}^{\mathsf{mult}[k,k]}$ give a “nearly” valid degree-$d=2k$ pseudomoment extension of $\bm{M}^{(\delta)}$. This is a more informal formulation of the expected correctness of the coupled tensor construction; the source does not specify a formal meaning for “nearly” or establish the claim.