Straight-condition conjecture for Hörmander-type operators

Let $n\geq 3$ and let $T^{\lambda}$ be a Hörmander-type operator satisfying the straight condition: for each $\xi$, the associated quantity $G(x,\xi)$ is invariant under changes in $x$. Straight-condition conjecture. For every $\varepsilon>0$, the estimate

$$\|T^{\lambda}f\|_{L^p(\mathbb R^n)}\lesssim_{\phi,a}\|f\|_{L^p(B^{n-1}_1(0))}$$

holds uniformly for $\lambda\geq 1$ whenever $p>\frac{2n}{n-1}$. This is proposed as a range in which the Kakeya compression obstruction is absent; the source does not establish whether the conjecture is open or resolved.