Let $X$ be a smooth projective variety and $E\longrightarrow X$ a vector bundle of rank $n$. For any $0\leq k\leq n$ the associated Grassmann bundle $G_k(E)\longrightarrow X$ yields and we have the so-called "basis theorem" (see Fulton "Intersection theory", Proposition 14.6.5) which asserts that for any $s\geq 0$, $$CH_s(G_k(E))=\bigoplus_{\lambda}CH_{s-k(n-k)+|\lambda|}(X)$$ where $\lambda$ runs over all partitions $\lambda=(\lambda_1,...,\lambda_k)$ with $n-k\geq \lambda_1\geq...\geq\lambda_k\geq 0$.

I would like to know if there is some similar result for fiber product of Grassman bundles, i.e. consider a vector bundle $E\longrightarrow X$ of rank $n$ and $0\leq k_1\leq k_2\leq n$ two integer : does the $s$-dimensional Chow group $CH_s(G_{k_1}(E)\times_{X} G_{k_2}(E))$ is isomorphic to a direct sum of $\bigoplus_{p\in \mathcal{P}}CH_p(X)$ for some set $\mathcal{P}$ ?

I tried to look a an answer considering the immersion $G_{k_1}(E)\times_X G_{k_2}(E)\longrightarrow G_{k_1k_2}(E\times_X E)$ without success.

This question studying the case where $X=\mathbb{P}^1$ and $E$ the tautological vector bundle.