Adding a pulldown was as far as I got. Unlike others I failed to note the value I used. Summary here. I imagine it was similar to @iampete’s value as that works…
Looking at the datasheet for the TXS0108E, the chip’s pullup is 4k when driving high. TI also have an application report for external pull resistors with TXS devices: Effects of External Pullup and Pulldown Resistors
on TXS and TXB Devices
TI recommend 50kOhms or greater, though I’m not sure if that will actually work in situ.