Parameter-family conjecture for polycyclic geometric $(7m_4)$ configurations

Let $m>3$, and consider the voltage group $\mathbb Z_m$. For parameters $a,b$, define the two parameter sets $F_1(m;a,b)$ and $F_2(m;a,b)$ by

$$F_1(m;a,b)=\{a,c,d,e,f,g,q,a',c',d',e',f',g',q',t\}=\{a,-a,a,b,b,-a,-2a,a,-a,a,b,b,-a,-2a,0\}$$

for $a\geq b$ and $1\leq a,b\leq m/2$, and

$$F_2(m;a,b)=\{a,c,d,e,f,g,q,a',c',d',e',f',g',q',t\}=\{a,b,b,b,b,b,b,a,b,b,b,b,b,b,0\}$$

for $a\geq b$ and $a\neq m/2$. Parameter-family conjecture. These parameter values lead to geometric $(7m_4)$ configurations realizable polycyclically over $\mathbb Z_m$, and there are other valid as-yet-unidentified parameter families as well. The claim is based on preliminary experiments and the source explicitly notes that further constraints on $a$ and $b$ may not yet have been identified; proving and understanding the resulting families is left for future research.