The partition polynomial convolution identity

Let $\Lambda=(\lambda_1,\dots,\lambda_d)$ be a finite sequence with sum

$$s(\Lambda)=\sum_{i=1}^d\lambda_i.$$

Define the polynomial $F_\Lambda(x)$ by $F_\emptyset(x)=1$ and, for $1\leq d=\sharp(\Lambda)<\infty$,

$$F_\Lambda(x)=x{x-1+s(\Lambda)\choose d-1}(d-1)!.$$

For a map $\varphi\colon\{1,\dots,d\}\to\{1,\dots,k\}$, let $\varphi^{-1}(1),\dots,\varphi^{-1}(k)$ be the induced sub-sequences of $\Lambda$. Partition polynomial convolution identity. For arbitrary $x_1,\dots,x_k$, one has

$$\sum_{\varphi\in\{1,\dots,k\}^{\{1,\dots,d\}}}\prod_{j=1}^k F_{\varphi^{-1}(j)}(x_j)=F_\Lambda\left(\sum_{j=1}^k x_j\right).$$

This is presented as an identity rather than an explicitly named conjecture; the supplied parser nevertheless identifies it as a conjecture environment, and no proof or resolution information is provided in the source excerpt.