The symmetric 4-term amicable-pair conjecture

Let $A=2^n abcd$ and $B=2^n efgh$ be a symmetric higher-order amicable pair, with $a,b,c,d,e,f,g,h$ denoting the relevant prime factors. Define $x$ and $y$ through the normalized expressions associated with $A$, $B$, and the sum $\varphi(A)+\varphi(B)$ described in the preceding results. Symmetric 4-term conjecture. The relationship between the symmetric polynomials in the prime factors is hypothesized to satisfy

$$y-1=[ef(g+h)+gh]-(a+b+c+d),$$ $$x-1=[ab(c+d)+cd]-(e+f+g+h).$$

This proposes a higher-order symmetric extension of the factorization patterns established earlier for amicable pairs whose greatest common divisor is a power of two.