The only maximal affine Hadamard families stemming from the $10$-dimensional Fourier matrix $F_{10}$ are $$F_{10}^{(4)}(a,b,c,d) = F_{10}\circ{\rm EXP}\left(i R_{F_{10}^{(4)}}(a,b,c,d)\right)$$ and $$\left(F_{10}^{(4)}(a,b,c,d)\right)^{\rm T} = F_{10}\circ{\rm EXP}\left(i\left(R_{F_{10}^{(4)}}(a,b,c,d)\right)^{\rm T}\right),$$ where $$F_{10}=\left[\begin{array}{llllllllll} 1& 1& 1& 1& 1& 1& 1& 1& 1& 1\\ 1& w&w^2&w^3&w^4&w^5&w^6&w^7&w^8&w^9\\ 1&w^2&w^4&w^6&w^8& 1&w^2&w^4&w^6&w^8\\ 1&w^3&w^6&w^9&w^2&w^5&w^8& w&w^4&w^7\\ 1&w^4&w^8&w^2&w^6& 1&w^4&w^8&w^2&w^6\\ 1&w^5& 1&w^5& 1&w^5& 1&w^5& 1&w^5\\ 1&w^6&w^2&w^8&w^4& 1&w^6&w^2&w^8&w^4\\ 1&w^7&w^4& w&w^8&w^5&w^2&w^9&w^6&w^3\\ 1&w^8&w^6&w^4&w^2& 1&w^8&w^6&w^4&w^2\\ 1&w^9&w^8&w^7&w^6&w^5&w^4&w^3&w^2& w \end{array}\right]:\quad w=\exp(2\pi i/10)$$ and $$R_{F_{10}^{(4)}}(a,b,c,d)=\left[\begin{array}{llllllllll} \bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet\\ \bullet&a&b&c&d&\bullet&a&b&c&d\\ \bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet\\ \bullet&a&b&c&d&\bullet&a&b&c&d\\ \bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet\\ \bullet&a&b&c&d&\bullet&a&b&c&d\\ \bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet\\ \bullet&a&b&c&d&\bullet&a&b&c&d\\ \bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet\\ \bullet&a&b&c&d&\bullet&a&b&c&d\end{array}\right].$$ The affine Hadamard families $F_{10}^{(4)}$ and $\left(F_{10}^{(4)}\right)^{\rm T}$ are cognate.