Ishii–Stade shift-equation conjecture for Mellin transforms

Let $m,n\in \mathbb Z$ with $1\le m\le n-1$ and let $\delta\in \mathbb Z_{\ge0}$. For $x\in\mathbb C$ and $q\in\mathbb Z_{\ge0}$, write

$$(x)_q=\frac{\Gamma(x+q)}{\Gamma(x)}=x(x+1)\cdots(x+q-1).$$

Let $\what{n}{\alpha}{s}$ be the Mellin transform under consideration, with $s=(s_1,\ldots,s_{n-1})$ and Langlands parameters $\alpha=(\alpha_1,\ldots,\alpha_n)$. Then there exist a positive integer $r$, polynomials $P_i(s,\alpha)$, and tuples $\Sigma_i\in(\mathbb Z_{\ge0})^{n-1}$ such that

$$\what{n}{\alpha}{s}=\left[\prod_{1\le j_1<j_2<\cdots<j_m\le n}\left(s_m+\alpha_{j_1}+\alpha_{j_2}+\cdots+\alpha_{j_m}\right)_\delta\right]^{-1}\sum_{i=1}^r P_i(s,\alpha)\what{n}{\alpha}{s+\Sigma_i},$$

where the $m$th coordinate of every $\Sigma_i$ is at least $\delta$, and

$$\deg(P_i(s,\alpha))+2|\Sigma_i|=\delta\binom{n}{m}$$

for every $i$. This conjecture supplies the shift equations needed to continue the Mellin transform beyond its initial domain; it is known for $2\le n\le7$, and for $m=1$ for every $n\ge2$, while the general case remains open.