The positive-curvature restriction conjecture under geometric control

Let $(M,g)$ be a Riemannian manifold and let $\boldsymbol{\Sigma}$ be an interior hypersurface with positive definite second fundamental form satisfying the transversal geometric control condition, denoted $\mathcal{T}$GCC. Let $(\lambda,\phi)$ satisfy the eigenfunction equation $(-\Delta_g-\lambda^2)\phi=0$. Controlled restriction conjecture. There exist $C,c,\lambda_0>0$ such that, for all $(\lambda,\phi)\in[\lambda_0,\infty)\times L^2(M)$,

$$\|\phi\|_{L^2(M)}\leq C\|\phi|_{\Sigma}\|_{L^2(\Sigma)},\qquad \|\phi\|_{L^2(M)}\leq C\|\lambda^{-1}\partial_\nu\phi|_{\Sigma}\|_{L^2(\Sigma)}.$$

The preceding theorem establishes the corresponding estimate using both traces under $\mathcal{T}$GCC, while this stronger pair of single-trace estimates is proposed in the source and remains open.