Stronger congruence conjecture for projective congruence data

Let a projective congruence datum that is also Galois have exponent $N$ and global dimension $n$. Stronger congruence conjecture. If $n\equiv 2\pmod{4}$, then

$$N\equiv 4\pmod{8}$$

and

$${\mathfrak g}^2=-t_o^4{\mathfrak g}'^2.$$

The conjecture strengthens a preceding result proving only $N\equiv0\pmod{4}$ under these assumptions. It concerns the relation between the exponent and global dimension of Galois projective congruence data.