Left properness conjecture for q-model structures on multipointed d-spaces

Take ${\mathbf{Top}}$ to be the category of $\Delta$-Hausdorff $\Delta$-generated spaces. Left properness conjecture. The q-model structures of ${\rm{\mathbf{{\mathcal{G}}dTop}}}$ and ${\rm{\mathbf{{\mathcal{M}}dTop^{sat}}}}$ are left proper. This conjecture concerns the homotopical behavior of q-model structures on multipointed d-spaces; left properness would ensure that pushouts along cofibrations preserve weak equivalences. The supplied text does not indicate whether the conjecture has been resolved.