Gordon's conjecture on connected sums of Heegaard splittings

Let $M_1=V_1\bowtie_{F_1}W_1$ and $M_2=V_2\bowtie_{F_2}W_2$ be Heegaard splittings, and call a Heegaard splitting unstabilized if it is not stabilized. Form their connected sum, written

$$(V_1\bowtie_{F_1}W_1)\#(V_2\bowtie_{F_2}W_2)=(V_1\natural V_2)\bowtie_{F_1\# F_2}(W_1\natural W_2).$$

Gordon's conjecture. The connected sum of two unstabilized Heegaard splittings is unstabilized.

This is the Heegaard-splitting statement underlying the bridge version discussed in the paper. The source states that Gordon's conjecture was proved by Bachman and independently by Qiu and Scharlemann.