Cartier–Manin matrix conjecture for the parameter $a$

Let $H_a$ be a genus-4 hyperelliptic curve, let $M_{H_a}$ be its Cartier–Manin matrix, and let $a$ denote the parameter defining the curve. Cartier–Manin matrix conjecture. The greatest common divisor of all entries of $M_{H_a}$ does not have $a=\pm 2$. The supplied text gives no surrounding explanation of the intended divisibility or parameter condition, so the precise mathematical meaning and status beyond its conjectural presentation require checking in the paper.