Betti numbers and shifts in minimal graded free resolutions

Let $S$ be a polynomial ring over a field $K$ and let $R=S / I$ be a standard graded $K$-Algebra where $I$ is a graded ideal of $S$. The Multiplicity Conjecture of Herzog, Huneke, and Srinivasan which was recently proved using the Boij-Söderberg theory states that the multiplicity of $R$ is bounded above by a function of the maximal shifts in the minimal graded free resolution of $R$ over $S$ as well as bounded below by a function of the minimal shifts if $R$ is Cohen-Macaulay.

We present results to the related problem that the total Betti-numbers of $R$ are also bounded above by a function of the shifts in the minimal graded free resolution of $R$ as well as bounded below by another function of the shifts if $R$ is Cohen-Macaulay.