Toric pullback irreducibility conjecture for finite maps

Let $W$ be an irreducible quasiprojective variety and let $\varphi\colon W\to\mathbb{G}_{\rm m}^n$ be a map finite onto its image. Let $\overline{\varphi(W)}$ denote the closure of its image.

Toric pullback irreducibility conjecture. There is a finite union $\mathcal{E}$ of proper subtori of $\mathbb{G}_{\rm m}^n$ and a finite set $\Lambda$ of isogenies of $\mathbb{G}_{\rm m}^n$ such that, for every subtorus $T\subset\mathbb{G}_{\rm m}^n$ satisfying

$$\dim(T)\geq\operatorname{codim}(\overline{\varphi(W)})+1$$

and every point $p\in\mathbb{G}_{\rm m}^n(\mathbb{C})$, one of the following holds: $T\subseteq\mathcal{E}$; there is $\lambda\in\Lambda$ such that $\lambda^*W$ is reducible and a subtorus $T'\subset\mathbb{G}_{\rm m}^n$ on which $\lambda$ induces an isomorphism $T'\to T$; or $\varphi^{-1}(p\cdot T)$ is irreducible.

This conjecturally extends the preceding toric Bertini statement by allowing reducibility to be detected after finitely many isogeny pullbacks. Its status is not resolved in the supplied text.