Characterization of the period and preperiod for three-element subtraction sets

Let $A=\{a,b,c\}$ with $1<a<b<c$. Define integers by the division algorithm:

$$b=qa+r,$$ $$c=q_c(a+b)+r_c,\qquad r_c=q_a(2a)+r_a,$$ $$c-a=q_c'(a+b)+r_c',\qquad r_c'=q_a'(2a)+r_a'.$$

Here $\operatorname{Per}(A)$ and $\operatorname{PrePer}(A)$ denote the eventual period and preperiod. Three-element period–preperiod conjecture. One has $\operatorname{Per}(A)=b+c$ and $\operatorname{PrePer}(A)=0$ if and only if at least one of the following holds:

1. $q$ is even, $r_c'>0$, $r_a'\leq r$, and $2q_a'\leq q$; if $2q_a'=q$, then $r_a'\leq 2r-a$.
2. $q$ is odd, $r\neq 0$, $r\leq r_a\leq a$; if $q_a=0$, then $r_a<a$.
3. $q$ is odd, $r=0$, and $r_a\neq a$.

The criterion is based on computer simulations for three-element sets up to a bounded range and is intended as a limited characterization of the complex general $\{a,b,c\}$ case; its status remains conjectural.