Las Vergnas's second Tutte polynomial conjecture for binary matroids

Let $M$ be a binary matroid, let $T_M(x,y)$ denote its Tutte polynomial, and let $z$ be an integer. Las Vergnas's second conjecture. The value

$$\frac{T_M(-1+4z,-1+4z)}{T_M(-1,-1)}$$

is an odd integer for every binary matroid $M$ and every integer $z$. This conjecture is the second of three conjectures made by Michel Las Vergnas concerning evaluations of the Tutte polynomial of binary matroids; it is stronger than the first conjecture and weaker than the third. The first conjecture was proved, while the third was disproved; this second conjecture is presented here as the subject of counterexamples.