TITLE: Genus 2 3-manifolds bounding only $X^4$ with $b_2(X^4)$ big?
QUESTION [9 upvotes]: The genus of a closed orientable  3-manifold $M^3$ is the minimum genus among all Heegaard splitting surfaces for $M$.  Every such 3-manifold bounds a compact 4-manifold.  Let $I(M)$ denote the minimum second Betti number $b_2$ amongst all such bounding $X^4$.  
Is there a sequence of genus 2 3-manifolds $M_n$ such that $I(M_n) \to \infty$.

REPLY [2 votes]: I think it follows that a Heegaard genus 2 manifold will bound a topological 4-manifold with $b_2\leq 2$ from a theorem of Steve Boyer. 
Boyer, Steven, Simply-connected 4-manifolds with a given boundary, Trans. Am. Math. Soc. 298, 331-357 (1986). ZBL0615.57008. MR0857447
I'm guessing that you're interested in the smooth category though?