Witt rings of \(G\)-forms and product of trace \(G\)-forms and quadratic forms (Q1403884)

Let \(k\) be a field of characteristic different from 2. For any field extension \(L/k\) the trace form is defined by \(q_L(x,y)= \text{trace}_{L/k}(x\cdot y)\). Now let \(L/k\) be a Galois extension with Galois group \(G\). Then \(q_L\) is also a \(G\)-form. Let \(W(k)\) be the Witt ring of \(k\) and let \(I(k)\subset W(k)\) be the fundamental ideal. Suppose \(cd_2(k)\leq d\). Eva Bayer-Fluckiger asked the following questions: Question 1. Does \(\varphi\in I^d(k)\) imply that \(\varphi \otimes q_L\) and \(\varphi\otimes q_{L'}\) are \(G\)-isomorphic? Question 2. Let \(\varphi\in I^{d-1}(k)\). Is is true that \(\varphi\otimes q_L\) and \(\varphi\otimes q_{L'}\) are \(G\)-isomorphic if and only if \(e_{d-1}(\varphi)\cup (x_L)=e_{d-1} (\varphi)\cup (x_{L'}) \in H^d(k,\mu_2)\) for all \(x\in H^1(G, \mu_2)\)? Here we use the usual notation of the Milnor-Voevodsky theorem. The author gives an affirmative answer to the first question and for the second question for \(d=2\).