The monotonicity conjecture for the spherical restriction functional

Let $d\geqslant 2$, let $q_d=2(d+1)/(d-1)$, and let $\Phi_{d,q}$ be the normalized Fourier restriction functional for the unit sphere. Define $\gamma_d(q)$ as in the source's inequality relating $\Phi_{d,q+2}(\mathbf{1})$ and $\Phi_{d,q}(\mathbf{1})$. Suppose $q_\star\geqslant q_d$ satisfies

$$\Phi_{d,q_\star+2}(\mathbf{1})\geqslant \gamma_d(q_\star)\Phi_{d,q_\star}(\mathbf{1}).$$

Monotonicity conjecture. Then

$$\Phi_{d,q+2}(\mathbf{1})>\gamma_d(q)\Phi_{d,q}(\mathbf{1})$$

for every $q>q_\star$.

The conjecture is intended to settle the remaining observation concerning the inequality preceding it and is presented as a question for future work; the source gives no proof or resolution.