Branched continued fraction for extended multivariate Eulerian polynomials of negative type

Let $m\geq 1$, let ${\mathbf{x}}=(x_0,\ldots,x_{m-1})$ and ${\mathbf{c}}=(c_L)_{L\geq 0}$ be indeterminates, and let ${\mathcal{P}}^{(m)-}_n({\mathbf{x}},{\mathbf{c}})$ be the extended multivariate Eulerian polynomial of negative type defined from increasing multi-$m$-ary trees with level weights. Let $\widehat{P}_n^{(m,m)}({\mathbf{x}},{\mathbf{c}})$ and $S_n^{(m)}({\bm{\alpha}})$ denote the corresponding branched continued-fraction polynomials, with weights ${\bm{\alpha}}$ given by the factorized formula with period $p=m$. Branched continued fraction conjecture, part (a). For every integer $m\geq 1$,

$${\mathcal{P}}^{(m)-}_n({\mathbf{x}},{\mathbf{c}})=\widehat{P}_n^{(m,m)}({\mathbf{x}},{\mathbf{c}})=S_n^{(m)}({\bm{\alpha}}).$$

The analogous identities for the other extended multivariate Eulerian polynomials are proved in the paper, but this part is not proved and is presented as the remaining conjectural case.