The even-degree extension of Langmann's irreducibility theorem

Let $H(t,X)\in\mathbb Q[t,X]$ be a homogeneous polynomial of degree $n$ that is not a proper power. Consider the specialization $H(\bar t,X)-1$ for integers $\bar t$. Even-degree irreducibility conjecture. The odd-degree hypothesis in the preceding irreducibility theorem should be removable provided that

(a) $n\ne 2,4$; and

(b) if $4$ divides $n$, then $-4H(t,X)$ is not a fourth power in $\mathbb Q[t,X]$.

Under these necessary conditions, $H(\bar t,X)-1$ should be irreducible for all but finitely many $\bar t\in\mathbb Z$. The stated conditions exclude the known obstructions in even degree: degree $2$ and degree $4$, together with the case in which $-4H(t,X)$ is a fourth power when $4$ divides $n$.