  For an immersed minimal surface in $\mathbb{R}^3$, we show that there exists
a lower bound on its Morse index that depends on the genus and number of ends,
counting multiplicity. This improves, in several ways, an estimate we
previously obtained bounding the genus and number of ends by the index. Our new
estimate resolves several conjectures made by J. Choe and D. Hoffman concerning
the classification of low-index minimal surfaces: we show that there are no
complete two-sided immersed minimal surfaces in $\mathbb{R}^3$ of index two,
complete embedded minimal surface with index three, or complete one-sided
minimal immersion with index one.
