Three lens space summands from the Poincar\'e homology sphere

A regular fiber of the Seifert fibering of the Poincar'e homology sphere admits a Dehn surgery to $L(2,1)\mathrm{\#}L(3,2)\mathrm{\#}L(5,4)$. We prove that this is the only knot in the Poincar'e homology sphere with a surgery to a connected sum of more than two lens spaces.