Conca–Krattenthaler–Watanabe's regular-sequence conjecture for four power sums

Let $A\subset\mathbb{N}^*$ have $|A|=4$, say $A=\{a_1,a_2,a_3,a_4\}$, with $\gcd(A)=1$, and let $p_A(4)$ denote the corresponding four power sum symmetric polynomials in four variables. Conca–Krattenthaler–Watanabe's conjecture. The sequence $p_A(4)$ is regular if and only if: at least two $a_i$ are even, at least one is divisible by $3$, and at least one is divisible by $4$; if $E$ is the set of even elements of $A$ and $d=\gcd(E)$, then $\{a/d:a\in E\}$ contains an even number; and $A$ does not contain a subset of the form $\{d,2d,5d\}$. The source presents this as the recalled conjecture on power sums in four variables; no resolution is supplied here.