Conjecture on the complete range of elliptic hyperboloid extension estimates

Let $d$ be the spatial dimension, let $\Sigma$ be the elliptic hyperboloid equipped with its Lorentz-invariant measure $\mu$, and let $\mathcal{E}$ denote the associated extension operator. For $p,q\in[1,\infty]$, write $\mathcal{E}(p\rightarrow q)$ when $\mathcal{E}$ extends boundedly from $L^p(\Sigma,\mu)$ to $L^q(\mathbb{R}\times\mathbb{R}^d)$. Complete-range extension conjecture. If

$$\left(\frac{d}{d+2}q\right)' \leq p \leq \min\left\{\left(\frac{d-1}{d+1}q\right)',q\right\}$$

and $(p,q)\neq\left(\frac{2d}{d-1},\frac{2d}{d-1}\right),\left(\frac{2(d+1)}{d},\frac{2(d+1)}{d}\right)$, then $\mathcal{E}(p\rightarrow q)$ holds. This conjecture seeks the complete range of bounded extension estimates for the elliptic hyperboloid, including the non-endpoint exponent region while excluding the two stated pairs; the source provides no resolution status for the conjecture.