Sun's general conjecture on linear and circular restricted sumsets

Let $G$ be an additive group with $|G|>1$. For finite subsets $A_1,\ldots,A_n$ of $G$, define

$$L(A_1,\ldots,A_n)=\{a_1+\cdots+a_n:a_i\in A_i\text{ for }i=1,\ldots,n,\text{ and }a_i\ne a_{i+1}\text{ for }0<i<n\}$$

and

$$C(A_1,\ldots,A_n)=\{a_1+\cdots+a_n:a_i\in A_i\text{ for }i=1,\ldots,n,\ a_i\ne a_{i+1}\text{ for }0<i<n,\text{ and }a_n\ne a_1\}.$$

Let $n>1$ and assume that $|A_i|>1$ for every $i=1,\ldots,n$. The Sun restricted-sumset conjecture.

$$|L(A_1,\ldots,A_n)|\geqslant\min\left\{p(G),\,|A_1|+\cdots+|A_n|-2n+1+\{n\}_2\right\}$$

and

$$|C(A_1,\ldots,A_n)|\geqslant\min\left\{p(G),\,|A_1|+\cdots+|A_n|-2n+(-1)^n(1+\{n\}_2)\right\},$$

where $p(G)$ is the minimum of the orders of the nonzero elements of $G$, interpreted as $+\infty$ if every nonzero element has infinite order, and $\{n\}_2$ denotes the least nonnegative residue of $n$ modulo $2$. This conjecture proposes general lower bounds for the two restricted sumsets introduced by Sun; the source does not state a resolution, so its status remains open.