Irreducibility conjecture for the four polynomial families $\widetilde C_{l,i,j}(X)$

Let $l\ge3$, and let $\widetilde C_{l,0,0}(X)$, $\widetilde C_{l,0,1}(X)$, $\widetilde C_{l,1,0}(X)$ and $\widetilde C_{l,1,1}(X)$ be the four polynomial families introduced in the paper. Four-family irreducibility conjecture. Each of these four polynomials is irreducible over ${\mathbb Q}$:

$$\widetilde C_{l,0,0}(X),\ \widetilde C_{l,0,1}(X),\ \widetilde C_{l,1,0}(X),\ \widetilde C_{l,1,1}(X).$$

The paper reports checks for $l\le350$ and says that the property is considered likely for all larger $l$, but provides no general proof in the supplied excerpt.