Given a multilinear polynomial \( q=\sum_{I\subseteq [n]} c_I\prod_{i\in I} a_i\in\mathbb{R}[a_1,\ldots,a_n] \), let \( T(q) \) be the family of its terms and let \( \deg t \) be the degree of a term \( t \). The polynomial\( \widetilde{q}=\sum_{t\in T(q)} ((1-a_{n+1})(b+\deg t)+a-b)t \in \mathbb{R}[a_1,\ldots,a_n,a_{n+1}] \) is then affiine as well. It is shown that under broad conditions for reals \( a \) and \( b \), if \( q\ge 0 \) whenever \( a_1\le a_2\le \cdots\le a_n\le 1 \), then \( \widetilde{q}\ge 0 \) whenever \( a_1\le a_2\le \cdots\le a_n\le a_{n+1}\le 1 \). This result implies potentially a step in a proof that the coefficient polynomials of positive degree of the power series in \( t \) of probabilistically weighted harmonic means of the quantities \( (1-x_1t),\ldots,(1-x_kt) \) are nonpositive whenever \( x_1,\ldots,x_k \) are nonnegative.