Dowling–Wilson top-heavy conjecture for subspace numbers

Let $E=\{v_1,\ldots,v_n\}$ be a spanning subset of a $d$-dimensional complex vector space $V$, and let $w_i(E)$ denote the number of $i$-dimensional subspaces spanned by subsets of $E$. Dowling–Wilson top-heavy conjecture. For every $i<d/2$, one has

$$w_i(E)\leq w_{d-i}(E).$$

This conjecture, proposed by Dowling and Wilson in 1974, is a special case of a more general conjecture for matroids. The statement is known in some low-dimensional cases, including $w_1(E)\leq w_2(E)$ when $d=3$, but its general status is not established here.