The pseudorandomness conjecture for nondegenerate linear maps

Let $m,d$ be natural numbers with $d\geqslant m+2$, and let $L:\mathbb{R}^d\to\mathbb{R}^m$ be a surjective linear map. Let $V^*_{\operatorname{degen}}(m,d)$ denote the degenerate locus, and let $\nu_{N,w}^{\gamma}$ be the pseudorandom majorant defined from the smooth sieve weight. Pseudorandomness conjecture. If $L\notin V^*_{\operatorname{degen}}(m,d)$, then there exist a value of $\gamma$ and a function $w:\mathbb{N}\to\mathbb{R}_{\geqslant 0}$ with $w(N)\to\infty$ as $N\to\infty$ such that $\nu_{N,w}^{\gamma}$ is $(L,w)$-pseudorandom. This would extend the proved pseudorandomness theorem from algebraic-coefficient maps to all nondegenerate surjective linear maps.