The celebrated Mason's conjecture states that the sequence of independent set numbers of any matroid is log-concave, and even ultra log-concave. The strong form of Mason's conjecture was independently solved by Anari, Liu, Oveis Gharan and Vinzant, and by Brändén and Huh. The weak form of Mason's conjecture was also generalized to a polynomial version by Dowling in 1980 by considering certain polynomial analogue of independent set numbers. Given a matroid $M=(E,\mathcal{I})$ with $|E|=n$, for any $0\le k\le r_M$ define \begin{eqnarray*} f_k(M)=\sum_{I\in \mathcal{I},\,|I|=k}\left(\prod_{x_i\in I} x_i\right). \end{eqnarray*} Dowling conjectured that: \begin{eqnarray*} f_k^2(M) \ge f_{k-1}(M)\,f_{k+1}(M) \end{eqnarray*} holds for any $0< k< r_M$, where the partial order between two polynomials $f,g\in \mathbb{R}[x_1,x_2,\ldots,x_n]$ is defined by $f\geq g$ if $f-g$ is a polynomial with nonnegative coefficients. It is clear that Dowling's polynomial conjecture implies the weak form of Mason's conjecture. In this paper we completely solve Dowling's polynomial conjecture by using the theory of Lorentzian polynomials. Moreover, we prove a stronger result. For any matroid $M$ of rank $r_M$, \begin{eqnarray*} f_l^p(M)\ge \left(1+\frac{1}{(p-1)l}\right)\left(1+\frac{2}{(p-1)l}\right)\cdots\left(1+\frac{p-1}{(p-1)l}\right)f_{l+1}^{p-1}(M)f_{l-p+1}(M) \end{eqnarray*} holds for any integers $p \ge 2$ and $p-1 \lt l \lt r_M$.