A family of complex Hadamard matrices found by F. Szöllősi [25] reads: $$D_{14}^{(5)}(a,b,c,d,e)=D_{15}\circ{\rm EXP}\left(i R_{D_{14}^{(5)}}(a,b,c,d,e)\right),$$ where
$$R_{D_{14}^{(5)}}(a,b,c,d,e)=\left[\begin{array}{rrrrrrrrrrrrrr} \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & \bullet & a& b& \bullet & c& \bullet & \bullet & \bullet & c& \bullet & b& a& \bullet\\ \bullet & -a& \bullet & \bullet & \bullet & c - a& e - a& \bullet & e - a& c - a& \bullet & \bullet & \bullet & -a\\ \bullet & -b& \bullet & \bullet & d - b& \bullet & e - b& \bullet & e - b& \bullet & d - b& \bullet & \bullet & -b\\ \bullet & \bullet & \bullet & b - d& \bullet & c - d& e - d& \bullet & e - d& c - d& \bullet & b - d& \bullet & \bullet\\ \bullet & -c& a - c& \bullet & d - c& \bullet & \bullet & \bullet & \bullet & \bullet & d - c& \bullet & a - c& -c\\ \bullet & \bullet & a - e& b - e& d - e& \bullet & \bullet & \bullet & \bullet & \bullet & d - e& b - e& a - e& \bullet\\ \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & \bullet & a - e& b - e& d - e& \bullet & \bullet & \bullet & \bullet & \bullet & d - e& b - e& a - e& \bullet\\ \bullet & -c& a - c& \bullet & d - c& \bullet & \bullet & \bullet & \bullet & \bullet & d - c& \bullet & a - c& -c\\ \bullet & \bullet & \bullet & b - d& \bullet & c - d& e - d& \bullet & e - d& c - d& \bullet & b - d& \bullet & \bullet\\ \bullet & -b& \bullet & \bullet & d - b& \bullet & e - b& \bullet & e - b& \bullet & d - b& \bullet & \bullet & -b\\ \bullet & -a& \bullet & \bullet & \bullet & c - a& e - a& \bullet & e - a& c - a& \bullet & \bullet & \bullet & -a\\ \bullet & \bullet & a& b& \bullet & c& \bullet & \bullet & \bullet & c& \bullet & b& a& \bullet \end{array}\right]$$ and $$D_{14} = \left[\begin{array}{rrrrrrrrrrrrrr} 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1\\ 1& -1& i& -i& i& i& -i& -i& -i& -i& i& i& -i& i\\ 1& i& -1& i& -i& i& i& -i& -i& -i& -i& i& i& -i\\ 1& -i& i& -1& i& -i& i& i& -i& -i& -i& -i& i& i\\ 1& i& -i& i& -1& i& -i& i& i& -i& -i& -i& -i& i\\ 1& i& i& -i& i& -1& i& -i& i& i& -i& -i& -i& -i\\ 1& -i& i& i& -i& i& -1& i& -i& i& i& -i& -i& -i\\ 1& -i& -i& i& i& -i& i& -1& i& -i& i& i& -i& -i\\ 1& -i& -i& -i& i& i& -i& i& -1& i& -i& i& i& -i\\ 1& -i& -i& -i& -i& i& i& -i& i& -1& i& -i& i& i\\ 1& i& -i& -i& -i& -i& i& i& -i& i& -1& i& -i& i\\ 1& i& i& -i& -i& -i& -i& i& i& -i& i& -1& i& -i\\ 1& -i& i& i& -i& -i& -i& -i& i& i& -i& i& -1& i\\ 1& i& -i& i& i& -i& -i& -i& -i& i& i& -i& i& -1 \end{array}\right].$$The defect $d\left(D_{14}\right)=36$, so it is not known whether $D_{14}^{(5)}$ is maximal affine, or further parameters can be introduced.