Polynomial-form conjecture for the hypergeometric Laplace transform

Let $x,y$ be vectors in the positive integration chamber, let $|x|$ denote the product of the coordinates of $x$, let $p$ be the measure parameter, let $e(-x,y)$ be the hypergeometric kernel, and let $x^\lambda$ denote the monomial indexed by the partition or multi-index $\lambda$. The associated integral is

$$\left(\int_0^\infty\right)^n e(-x,y)|x|^{a-p}x^\lambda\,\mathrm{d}\mu(x).$$

Polynomial-form conjecture. This integral is of the form

$$|y|^{-a}\times\text{a polynomial in }y^{-1}$$

with leading term $y^\lambda$. This is presented as a weaker form of Conjecture (C), whose full Laplace-transform identity would imply further hypergeometric integral formulas. The source does not provide a resolution of this weaker conjecture.