I know Qantas runs a shuttle bus for airside transfers from T4 to TBIT at LAX, but what about going in the other direction?

I'm booked on QF108 JFK-LAX (arriving at TBIT) connecting to QF26 LAX-AKL (departing from T4), and was wondering if there is any possibility for an airside transfer from TBIT to T4. as an alternative to exiting TBIT, walking over to T4 and going thru security there.

Thanks.