For each integer $k \geq 2$, Johnson gave a $3$-manifold with Heegaard splittings of genera $2k$ and $2k-1$ such that any common stabilization of these two surfaces has genus at least $3k-1$. We modify his argument to produce a $3$-manifold with two Heegaard splitings of genus $2k$ such that any common stabilization of them has genus at least $3k$.
"A refinement of Johnson's bounding for the stable genera of Heegaard splittings." Osaka J. Math. 48 (1) 251 - 268, March 2011.