6 The vertex of a circular cone is $P$, and its axial section is an isosceles right triangle. $A$ is a point on the circumference of the base circle, $B$ is a point inside the base circle, $O$ is the center of the base circle, $A B \perp$ $O B$, with the foot of the perpendicular at $B, O H \perp P B$, with the foot of the perpendicular at $H$, and $P A=4, C$ is the midpoint of $P A$. Then, when the volume of the tetrahedron $O-H P C$ is maximized, the length of $O B$ is ().
(A) $\frac{\sqrt{5}}{3}$
(B) $\frac{2 \sqrt{5}}{3}$
(C) $\frac{\sqrt{6}}{3}$
(D) $\frac{2 \sqrt{6}}{3}$