I would like to know if there is a polyhedral analog to this beautiful theorem of Hong:

Theorem 11.0.1.Any smooth positive disk $(\bar{D},g)$ with a positive geodesic curvature along $\partial D$ admits a smooth isometric embedding in $\mathbb{R}^3$ that maps $\partial D$ to a planar curve. Moreover, the embedding is unique within rigid motions.

This is proved in the book by
Qing Han and Jia-Xing Hong,
*Isometric Embedding of Riemannian Manifolds in Euclidean Spaces*,
American Mathematical Society Monograph, Volume 130, 2006, p.225.

I would like to replace the disk by a convex polyhedral cap $C$
with a convex boundary $\partial C$. $C$ is a connected subset of the
surface
of a convex polyhedron, and $\partial C$ has at every point $p \in \partial C$ at most $\pi$ angle
incident to $p$ within $C$:

Alexandrov's book
Convex Polyhedra
includes an article in the Appendix written by L.A. Shor
"On Flexibility of Convex Polyhedra with Boundary,"
which describes (rather complex) conditions under which flexing can
occur,
but does not seem to address flexing to achieve planarity of $\partial C$.
It appears that Alexandrov's and Shor's results imply that
it cannot always be possible, because (for example) if $\partial C$
contains no vertices of $C$, then it must be rigid
(but not necessarily planar).

But perhaps there are conditions under which some polyhedral analog of Hong's theorem holds? Thanks for pointers!

**Addendum.** My concentration on the conditions for "flexing" above was misplaced,
as Sergei Ivanov's answer demonstrates: the analog isometrically embeds a convex cap $C$ with
$\partial C$ planar, but not by hinging $C$'s faces as rigid plates,
but rather through an embedding which
in general alters the facial structure of $C$ while maintaining isometry.

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