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Toric topology assigns to each $n$-dimensional combinatorial simple convexpolytope $P$ with $m$ facets an $(m+n)$-dimensional moment-angle manifold$\mathcal{Z}_P$ with an action of a compact torus $T^m$ such that$\mathcal{Z}_P/T^m$ is a convex polytope of combinatorial type $P$. We studythe notion of $B$-rigidity. A property of a polytope $P$ is called $B$-rigid,if any isomorphism of graded rings $H^*(\mathcal{Z}_P,\mathbb Z)=H^*(\mathcal{Z}_Q,\mathbb Z)$ for a simple $n$-polytope $Q$ implies that italso has this property. We study families of $3$-dimensional polytopes definedby their cyclic $k$-edge-connectivity. These families include flag polytopesand Pogorelov polytopes, that is polytopes realizable as bounded right-angledpolytopes in Lobachevsky space $\mathbb L^3$. Pogorelov polytopes includefullerenes -- simple polytopes with only pentagonal and hexagonal faces. It isknown that the properties to be flag and Pogorelov polytope are $B$-rigid. Wefocus on almost Pogorelov polytopes, which are strongly cyclically$4$-edge-connected polytopes. They correspond to right-angled polytopes offinite volume in $\mathbb L^3$. There is a subfamily of ideal almost Pogorelovpolytopes corresponding to ideal right-angled polytopes. We prove that theproperties to be an almost Pogorelov polytope and an ideal almost Pogorelovpolytope are $B$-rigid. As a corollary we obtain that $3$-dimensionalassociahedron $As^3$ and permutohedron $Pe^3$ are $B$-rigid. We generalizemethods known for Pogorelov polytopes. We obtain results on $B$-rigidity ofsubsets in $H^*(\mathcal{Z}_P,\mathbb Z)$ and prove an analog of the so-calledseparable circuit condition (SCC). As an example we consider the ring$H^*(\mathcal{Z}_{As^3},\mathbb Z)$.

$B$-rigidity of the property to be an almost Pogorelov polytope