Chow-class coefficient conjecture for Schubert varieties of subspace arrangements

Let $Y_V \subset \mathbb{P}^{\mathbf n}$ be a Schubert variety of a subspace arrangement, and write its Chow class as

$$[Y_V] = \sum_{\mathbf b} c_{\mathbf b}[\mathbb{P}^{\mathbf b}].$$

For a basis $\mathbf b=(b_1,\ldots,b_N)$, Chow-class coefficient conjecture. The coefficient of $[\mathbb{P}^{\mathbf b}]$ in $[Y_V]$ is

$$c_{\mathbf b}=\binom{n_1}{b_1}\binom{n_2}{b_2}\cdots\binom{n_N}{b_N}.$$

The Chow class need not be multiplicity-free, and its coefficients may depend on $\mathbf n$. The conjectured formula gives an explicit combinatorial expression for the coefficients corresponding to basis vectors; it is presented as open in the source.