The Borel partition spectrum at successors of singular cardinals

Assuming that $0^{†}$ does not exist, we prove that if there is a partition of $\mathbb{R}$ into ${\mathrm{\aleph }}_{\omega }$ Borel sets, then there is also a partition of $\mathbb{R}$ into ${\mathrm{\aleph }}_{\omega +1}$ Borel sets.