A property of $\gamma$ curves derived from topology of relative trisected 4-manifolds

Any two $(g;0,0,k)$-trisected closed 4-manifolds are related to each other by cutting and regluing in regard to $\gamma$ curves by an element of the Torelli group. Moreover, Lambert-Cole showed that this element can be replaced by an element of the Johnson kernel. We consider similar situation in the relative case. Furthermore, we relate this situation to cork twist.