at.algebraic topology - Killing Chern classes

Let $G$ be a compact connected Lie group and let $Eto B$ be a principal $G$-bundle. Suppose $a$ is a rational cohomology class of $E$ such that its pullback $b$ under an orbit inclusion map $Gto E$ is one of the standard multiplicative generators of $H^{{bullet}}(G,mathbf{Q})$ . Let $E'=Etimes EG/G$ be the Borel construction (corresponding to the action of $G$ on $E$) and let $(E^{pq}_r,d_r)$ be the Leray spectral sequence corresponding to the fiber bundle $E'to BG$.

The class $a$ gives an element $a'in E^{0,2k-1}_2$ for some $k$. Assume that $d_i(a')=0,i< 2k$. Is it true that $d_{2k}(a')$ is what has remained in $E_{2k}$ of the multiplicative generator of $H^{{bullet}}(BG,mathbf{Q})$ corresponding to $b$?

For simplicity one can assume $G=U(n)$, in which case what remains in $E_infty$ of the generator of $H^{{{bullet}}}(BG,mathbf{Q})cong E^{{bullet},0}_2$ corresponding to $b$ is precisely the $k$-th Chern class of $E$, under the natural isomorphism $H^{{bullet}}(E',mathbf{Q})cong H^{{bullet}}(B,mathbf{Q})$.

This is probably standard, but for some reason I don't see how to prove it nor can construct a counter-example off hand.

upd: here is a weaker version, which would be easier to (dis)prove: take $G=U(n)times H$ where $H$ is another Lie group and suppose that the pullback of $a$ to $G$ is the canonical generator of $H^{bullet}(U(n),mathbf{Q})subset H^{bullet}(G,mathbf{Q})$ in degree $2k-1$. Is it true that $d_{2k}(a')$ is mapped to zero under the mapping of the spectral sequences induced by the pullback of $E'$ to $BH$ i.e. by the map

$$(Etimes EG)/Hto (Etimes EG)/G$$

To prove this it would suffice, of course, to show that $d_{2k}(a')$ is represented in $E_2$ by a class in $$H^{bullet}(BG,mathbf{Q})cong H^{bullet}(BU(n),mathbf{Q})otimes H^{bullet}(H,mathbf{Q})$$ that is mapped to zero under $H^{bullet}(BG,mathbf{Q})to H^{bullet}(BH,mathbf{Q})$.