Simplices with fixed volumes of codimension 2 faces in a continuous deformation

For any $n$-dimensional simplex in the Euclidean space ${\mathbb{R}}^n$ with $n\ge 4$, it is asked that if a continuous deformation preserves the volumes of all the codimension 2 faces, then is it necessarily a \emph{rigid} motion. While the question remains open and the general belief is that the answer is affirmative, for all $n\ge 4$, we provide counterexamples to a variant of the question where ${\mathbb{R}}^n$ is replaced by a pseudo-Euclidean space ${\mathbb{R}}^{p,n-p}$ for some unspecified $p\ge 2$.