The transcendental-coefficient conjecture for linear forms in primes

Let $L$, $\mathbf{v}$, $F$, and $G$ be as in Theorem 1, without assuming that $L$ has algebraic coefficients. Let $W=\prod_{p\leq w}p$. Transcendental-case conjecture. There should exist a function $w:\mathbb{N}\to\mathbb{R}_{\geqslant 0}$ with $w(N)\to\infty$ as $N\to\infty$ such that the conclusion of Theorem 1 holds with this choice of $W$. This conjecture seeks to remove the algebraicity restriction on the coefficients of $L$.