Let $X$ be a compact smooth 2-dimensional Riemannian manifold with boundary. Assume that the Gauss curvature of $X$ is at least $\kappa$, the diameter is at most $D$, and the second fundamental form of the boundary is at least $\lambda$.
Does there exist an upper bound on the area of $X$ in terms of $\kappa, D,\lambda$ only?
Remarks. (1) In the case of the empty boundary the answer is known to be positive (e.g. it follows from the Bishop inequality).
(2) If the boundary is non-empty and $\lambda\geq 0$, then the answer is also positive. In this case $X$ is locally geodesically convex and hence it is an Alexandrov space of curvature at most $\kappa$ in the sense of Alexandrov, and for such spaces there is a generalization of the Bishop inequality due to Burago-Gromov-Perelman. (Although there is a little bit more direct but longer explanation.)