The conjecture holds for $M=3,5,7$ but fails for $M=9$ when the resulting polynomial $\frac{16}3 k^4 - 84 k^3 + \frac{1334}3 k^2 - 896 k + 525$ is irreducible.
In fact, the polynomial is irreducible for all odd $M$ in the interval $[9,201]$.
Max Alekseyev
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