The high-density $3k-4$ conjecture modulo a prime

Let $G=\mathbb Z/p\mathbb Z$ with $p\geq 2$ prime, let $A,B\subseteq G$ be nonempty subsets with $A+B\neq G$, and set $C=-G\setminus(A+B)$. Suppose $|A|\geq |B|$ and

$$|A+B|=|A|+|B|+r\leq \min\{|A|+2|B|-3-\delta_B,\;p-r-3-\delta_C\},$$

where, for $X\in\{A,B,C\}$ with $(X,Y,Z)$ a permutation of $(A,B,C)$,

$$\delta_X=\begin{cases}1 & \text{if $r\geq 0$ and $X$ is a translate of either $Y$ or $Z$},\\1 & \text{if $r\geq 2$, $Y$ and $Z$ are translates of each other, and $|Y|=|Z|=r+4$},\\0 & \text{otherwise.}\end{cases}$$

The high-density $3k-4$ conjecture. There are arithmetic progressions $P_A,P_B,P_C\subseteq G$, all having a common difference, such that

$$X\subseteq P_X\quad\text{and}\quad |P_X|\leq |X|+r+1\quad\text{for all $X\in\{A,B,C\}$}.$$

This conjecture seeks a sharp extension of the integer $3k-4$ theorem to sumsets in $\mathbb Z/p\mathbb Z$ when the sumset and its complement satisfy the stated high-density bounds. The parser supplies no evidence resolving it, so its status is recorded as open.