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This suggests that, the result ought to be true under the weaker assumption that the faces of dimension $\leq 2$ have the same volumes.

Thinking about this I realize that the result is a consequence of H. Weyl's theorem on the invariants of the orthogonal group; see Theorem 9.3.7 of this book. In fact Weyl's result or the above proof show that the following stronger result is true.

Let $k\leq n$ be positive integers. If $(v_1,\dotsc, v_k)$ and $(w_1, \dotsc, w_k)$ are $k$-tuples of linearly independent vectors of $\bR^n$ such that $$ \big\vert(v_i,v_j)\big\vert=\big\vert(w_i,w_j)\big\vert,\;\;\forall i,j\in \{1,\dotsc , k\},$$ then there exists $T\in O(n)$ such that $Tv_i=\pm w_i$, $\forall i\in \{1,\dotsc, k\}$.

This suggests that, the result ought to be true under weaker assumptions.

Let $k\leq n$ be positive integers. If $(v_1,\dotsc, v_k)$ and $(w_1, \dotsc, w_k)$ are $k$-tuples of linearly independent vectors of $\bR^n$ such that $$ (v_i,v_j)=(w_i,w_j),\;\;\forall i,j\in \{1,\dotsc , k\},$$ then there exists $T\in O(n)$ such that $Tv_i=\pm w_i$, $\forall i\in \{1,\dotsc, k\}$.

Let $k\leq n$ be positive integers. If $(v_1,\dotsc, v_k)$ and $(w_1, \dotsc, w_k)$ are $k$-tuples of linearly independent vectors of $\bR^n$ such that $$ \big\vert(v_i,v_j)\big\vert=\big\vert(w_i,w_j)\big\vert,\;\;\forall i,j\in \{1,\dotsc , k\},$$ then there exists $T\in O(n)$ such that $Tv_i=\pm w_i$, $\forall i\in \{1,\dotsc, k\}$.

Define two equivalence relations"$\sim_n$" and "$\approx_n$" on the set of of bases of $\bR^n$.

$$ (v_1,\dotsc, v_n)\sim_n (w_1, \dotsc, w_n) $$

Let $k\leq n$ be positive integers. If $(v_1,\dotsc, v_k)$ and $(w_1, \dotsc, w_k)$ are $k$-tuples of linearly independent vectors of $\bR^n$ such that $$ \vol(v_i,\;i\in I)=\vol(w_i, \;i\in I), $$ for any $I\subset \{1,\dotsc, k\}$, $|I|\leq 2$, then there exists $T\in O(n)$ such that $Tv_i=\pm w_i$, $\forall i\in \{1,\dotsc, k\}$.

Define two equivalence relations"$\sim_n$" and "$\approx_n$" on the set of bases of $\bR^n$.

$$ (v_1,\dotsc, v_n)\sim_n (w_1,\dotsc, w_n) $$

Let $k\leq n$ be positive integers. If $(v_1,\dotsc, v_k)$ and $(w_1, \dotsc, w_k)$ are $k$-tuples of linearly independent vectors of $\bR^n$ such that $$ (v_i,v_j)=(w_i,w_j),\;\;\forall i,j\in \{1,\dotsc , k\},$$ then there exists $T\in O(n)$ such that $Tv_i=\pm w_i$, $\forall i\in \{1,\dotsc, k\}$.