Say a representation $\operatorname{Gal}(\mathbb Q) \to GL_n(\overline{\mathbb Q}_\ell)$ has big monodromy if the Zariski closure of the image of $\operatorname{Gal}(\mathbb Q) $ contains $SO_n$ or $Sp_n$.

Let $a_1,\dots a_n$ be natural numbers with $a_{n+1-i} =w-a_i$. Assume that

(1) some natural number that isn't $w/2$ appears twice among the $a_i$, and

(2) some gap appears among the $a_i$.

Do there necessarily exist infinitely many representations $\rho: \operatorname{Gal}(\mathbb Q) \to GL_n(\overline{\mathbb Q}_\ell)$, that are finitely ramified, de Rham at $\ell$ with Hodge-Tate weights $a_1,\dots,a_n$, and with big monodromy, up to twisting by one-dimensional characters?

Do there necessarily exist finitely many such representations?

The idea of the conditions is that (1) should prevent one from proving that there are infinitely many by automorphic methods, as the associated automorphic forms will not be discrete series (I think...) and (2) should prevent one from proving that there are infinitely many by geometric methods, as Griffiths transversality will prevent a family of such representations with big monodromy.

The "up to twisting by one-dimensional characters" prevents finding infintely many by trivial methods and the big monodromy prevents finding many automorphic forms on a different group and applying some representation with multiplicity in its weight spaces.

With these tools removed, are there still infinitely many representations? Or are there finitely many?