The four-term sum-product conjecture for fields of characteristic different from 2 and 3

Let $F$ be a field with ${\rm ch}(F)\not=2,3$. For any $x\in F$, choose elements $a,b,c,d\in F$.

Four-term sum-product conjecture. There are $a,b,c,d\in F$ such that

$$a+b+c+d-1=x=abcd.$$

This conjecture is motivated by the preceding representation theorem for an even number of summands and computational evidence; its validity for all fields of characteristic different from $2$ and $3$ remains open.