Classification conjecture for finite filled groups

Let $G$ be a finite group. A subset $S\subseteq G$ is product-free if $S\cap SS=\emptyset$, and it is complete if $G^{\ast}\subseteq S\cup SS$, where $G^{\ast}=G\setminus\{1\}$. A product-free set is locally maximal if it is not contained in a strictly larger product-free set. The group $G$ is filled if every locally maximal product-free subset of $G$ is complete. Write $C_m$ for the cyclic group of order $m$, $D_m$ for the dihedral group of order $m$, and $Q_8$ for the quaternion group of order $8$; $D_8\ast Q_8$ denotes their central product.

Classification conjecture. The group $G$ is filled if and only if $G$ is either an elementary abelian $2$-group or one of

$$C_3,\ C_5,\ D_6,\ D_8,\ D_{10},\ D_{12},\ D_{14},\ D_8\times C_2,\ D_{22},\ D_8\ast Q_8,\ (D_8\ast Q_8)\times C_2.$$

The authors report that this list agrees with computations in the small groups library through order $2000$, but the stated classification is presented as a conjecture and no resolution is supplied in the given text.