Sharp fully nonlinear Sobolev trace inequality on the ball

Let $(B^{n+1},dx^2)$ be the unit ball in Euclidean $(n+1)$-space. For a positive integer $k<\frac{n+1}{2}$, define

$$\mathcal{C}_{k-1}:=\left\{g=u^2dx^2\mid \sigma_j^g\geq 0,\ H_j^g>0,\ 1\leq j\leq k-1\right\}.$$

For $g\in\mathcal{C}_{k-1}$, let

$$\mathcal{S}_k(g):=\int_{B^{n+1}}\sigma_k^g\,\operatorname{dvol}_g+\oint_{S^n}H_k^g\,\operatorname{dvol}_{\iota^*g}.$$

Sharp Sobolev trace inequality. One has

$$\mathcal{S}_k(g)\geq \frac{n!}{(n+1-k)!(2k-1)!!}\omega_n^{\frac{2k-1}{n}}\left(\operatorname{Vol}_{\iota^*g}(S^n)\right)^{\frac{n+1-2k}{n}},$$

with equality if and only if $g$ is flat.

This is the expected fully nonlinear analogue of the sharp Sobolev trace inequality, extending results for the scalar-curvature case and related conformal curvature problems. The supplied context does not state whether the conjecture has been resolved.