Something no-one cares about enough to study, but the reason automatic faucets have never worked is because they have to detect a paw in the same place they spray water.  There's no way to know, once they get going, if the paw is still there or if it's just water.  They might have lowpass filters & try to detect a subtle decrease in reflectivity of a paw being removed, but this has never worked. 

The easiest solution is to have water spray for a preset amount of time, turn off before the next sensor reading, & start up again.  This might be what some of them did, but still failed at because the instinctive reaction of an animal is to remove his paws & try again when the water stops.  Another idea is to have them spray like a machine gun, taking rapid sensor readings. 

All proximity sensors are vulnerable to their threshold values rising over time & eventually shutting off when a paw is present. Decreasing the frequency of the sensor readings could require the ambient value to be more responsive to the fewer times it's sensing & be more prone to prematurely shutting off.

They're supposed to oscillate at 38khz to cancel out ambient light, but the on threshold is going to constantly adjust to the expected reflection of a paw.

The final solution may end up being a waterproof camera which just identifies a paw.  There's also adding more sensors elsewhere in the sink, but this gets expensive.

Paper towel dispensers have an easier time.  If something is there, don't dispense.  If nothing is there, dispense.  Some of them are designed to only dispense when a paw is there & these always fail.  The idea is animals take less product if the product isn't in a default state of being dispensed.  The trick is they have to go from a state of nothing present to something present before dispensing, then wait for something present to go to nothing present to reset.  The nothing present state has to last long enough to reset it.  If the animal puts his paw back in too soon, it won't detect the nothing present state.

This is a job for Technology Connections.