Consider a binary random vector $X=(X_1,\ldots,X_n)$ with a strong Rayleigh distribution (i.e., its multi-affine generating polynomial is stable). It is well known that the law of $X$ remains strong Rayleigh when conditioned on the value of $X_1$, or on the value of the total sum $X_1+\cdots+X_n$.

My question is: does this extend to the case where we condition on the value of the partial sum $X_1+\cdots+X_k$, for any $1\le k\le n$?