The completeness conjecture for MacMahon prime-detecting expressions

Let $M_a(n)$ be the partition functions defined by

$$M_a(n)=\sum_{\substack{0<s_1<s_2<\dots<s_a\\ n=m_1s_1+\dots+m_as_a}}m_1\cdots m_a.$$

Let $\vec{P}(x):=(p_1(x),p_2(x),\dots,p_a(x))\in\mathbb{Z}[x]^a$ be a vector of relatively prime integer polynomials. For integers $n\geq 2$, suppose that

$$E(n)=p_1(n)M_1(n)+p_2(n)M_2(n)+\dots+p_a(n)M_a(n)\geq 0,$$

and that $E(n)$ vanishes precisely when $n$ is prime. The completeness conjecture. Then $E(n)$ is a $\mathbb{Q}[n]$-linear combination of the entries in the table of known prime-detecting expressions. This conjectures that the listed MacMahon partition-function expressions generate all prime-detecting expressions of this type; the paper presents the claim as motivated by computer evidence, with no resolution supplied.