The ordered-partition expansion conjecture for basic sequences

Let $k$ be a field, let $\{\mathbf e_i\}_{1\leq i\leq \ell}$ be the standard basis of $k^\ell$, and let $\mathbf B=(B_1,\ldots,B_\ell)$ be a delta $\ell$-tuple with $B_i=D_iP_i^{-1}$. Let $b_{\mathbf n}(\mathbf x)$ be the basic sequence from the Transfer Theorem, let $B$ range over the ordered partitions of $[\ell]$, and let $\beta$ range over the partitions of $B$. With $\theta_j$ the umbral shift operator $p\mapsto x_jp$, write $\theta_\beta P_\beta$ for the corresponding products and $|B|$ for the number of parts of $B$. Ordered-partition expansion conjecture. The basic sequence from the Transfer Theorem can also be calculated as

$$b_{\mathbf n}(\mathbf x)=\sum\limits_{B\vdash[\ell]}(-1)^{\ell-|B|}\left(\theta_{\beta}P_{\beta}\right)_B\dfrac{\mathbf x^{\mathbf n-\mathbf 1}}{\mathbf n!}.$$

This conjectural expansion is intended to provide an alternative way to expand the Jacobian expression in the Transfer Theorem; the source gives it as an open question based on examples, and no general proof or resolution is supplied.