  In this short note, we formulate three problems relating to nonnegative
scalar curvature (NNSC) fill-ins. Loosely speaking, the first two problems
focus on: When are $(n-1)$-dimensional Bartnik data $\big(\Sigma_i ^{n-1},
\gamma_i, H_i\big)$, $i=1,2$, NNSC-cobordant? (i.e., there is an
$n$-dimensional compact Riemannian manifold $\big(\Omega^n, g\big)$ with scalar
curvature $R(g)\geq 0$ and the boundary $\partial \Omega=\Sigma_{1} \cup
\Sigma_{2}$ such that $\gamma_i$ is the metric on $\Sigma_i ^{n-1}$ induced by
$g$, and $H_i$ is the mean curvature of $\Sigma_i$ in $\big(\Omega^n, g\big)$).
If $\big(\mathbb{S}^{n-1},\gamma_{\rm std},0\big)$ is positive scalar curvature
(PSC) cobordant to $\big(\Sigma_1 ^{n-1}, \gamma_1, H_1\big)$, where
$\big(\mathbb{S}^{n-1}, \gamma_{\rm std}\big)$ denotes the standard round unit
sphere then $\big(\Sigma_1 ^{n-1}, \gamma_1, H_1\big)$ admits an NNSC fill-in.
Just as Gromov's conjecture is connected with positive mass theorem, our
problems are connected with Penrose inequality, at least in the case of $n=3$.
Our third problem is on $\Lambda\big(\Sigma^{n-1}, \gamma\big)$ defined below.
