Gouvêa's generalization of Mazur's dimension conjecture

Let $p$ be a prime, $K$ a number field, $S$ a finite set of primes of $K$, and $\overline{\rho}:G_{K,S}\to \operatorname{GL}_{n}(\mathbb{F})$ a continuous absolutely irreducible representation over a finite field $\mathbb{F}$ of characteristic $p$. Let $R_{\overline{\rho}}$ be its universal deformation ring, let $\operatorname{ad}:=\operatorname{ad}(\overline{\rho})$, and write $h^{i}(-):=\dim_{\mathbb{F}}H^{i}(-)$. Gouvêa's generalization. Mazur's dimension conjecture should hold without the condition that $S$ contain the primes above $p$ and the archimedean primes. This extends Mazur's proposed dimension formula to arbitrary finite sets of primes; the source cites Gouvêa's proposal and gives no resolution of the generalization.