  Given a Heegaard splitting of the $3$-sphere, the primitive disk complex is
defined to be the full subcomplex of the disk complex for one of the
handlebodies of the splitting. It is an open question whether the primitive
disk complex is connected or not when the genus of the splitting is greater
than three. In this note, we prove that a quotient of the primitive disk
complex, called the homotopy primitive disk complex, is connected.
