Exponential growth conjecture for compactly supported cohomology of moduli spaces of curves

Let $g$ be the genus of a smooth proper curve and let $k$ be a non-negative integer. Write $H^{2g+k}_c(\mathcal{M}_g)$ for the compactly supported cohomology of the moduli space $\mathcal{M}_g$ of smooth genus-$g$ curves. Exponential growth conjecture. The dimension of

$$H^{2g+k}_c(\mathcal{M}_g)$$

grows at least exponentially with $g$ for all but finitely many non-negative integers $k$. This would extend the known exponential-growth results, which currently cover every fixed $k$ with $0\leq k\leq 53$ except possibly $k\in\{1,4,7,20,51\}$; the finitely many exceptional values outside this range remain unspecified.