Samsung Galaxy Note9 rumors flip-flopped between yes and no on whether the late 2018 flagship will have an under display fingerprint reader. Things seem to have settled on no so the rumor mill has moved on to the Galaxy S10.
Current under display fingerprint readers work only with OLED displays since they are optical devices. And Samsung is the king of OLED when it comes to phones. However, rumor has it that the company is working on an in-house ultrasonic reader instead.
The advantages of such a sensor would be improved accuracy as it captures your finger in 3D while the other sensors are 2D only. However, ultrasonic readers are trickier to manufacture reliably, not to mention pricier.
Qualcomm has been promising ultrasonic readers for a while but none has reached the mainstream. vivo experimented with Qualcomms solution but ultimately went with Synaptics for the X21 UD.