A preordered field \((K,S)\) is said to satisfy the Weak Hilbert Property (WHP for short) if any rational function \(f(X_1,\cdots, X_m)\) over \(K\), which is positive definite on \(K\) for every ordering of \((K,S)\), can be represented in the form \(\sum s_ih_i^2(X_1,\cdots, X_m)\), where \(s_i \in S, h_i(X_1,\cdots, X_m)\in K(X_1,\cdots, X_m)\) for each \(i\). Let \((K^*,S^*)\) be an extension of \((K,S)\). The author continues his interest in a connection between two statements: ``\((K,S)\) satisfies WHP'' and ``\((K^*,S^*)\) satisfies WHP''. In his previous paper [Math. Z. 206, 145-151 (1991; Zbl 0691.12009)] he proved that if \((K,S)\) satisfies WHP and \((K^*,S^*)\) is a finitely generated extension of \((K,S)\), then \((K^*,S^*)\) also satisfies WHP. In the paper under review, the author constructs a counterexample to disprove the converse. The construction bases on the main result of the paper that says that if \((K^*,S^*)\) is a finitely generated extension of \((K,S)\) and \(K^*\) is transcendental over \(K\), then \((K^*,S^*)\) satisfies WHP (even if \((K,S)\) does not). Finally, the author proves that when \(S\) is an ordering and \(K^*\) is a finite extension of \(K\), then \((K^*,S^*)\) satisfies WHP if and only if \((K,S)\) satisfies WHP.