Should neutrals be tied together or left separate?
No.

The UPS includes that connection when required internally. That's why there is a "UPS Neutral Out"

UPS --> SPS

Most consumer products labeled "UPS" are actually SPS systems. UPS (Uninterruptable Power Supply) means that the nominal power path includes the battery output. That way power comes from the UPS battery power path regardless of whether there is any AC input or not... hence output is uninterruptable. 

Consumer units tend to be Standby Power Supplies where the output is switched (quickly) to the battery path when mainline AC becomes unavailable.

The role of neutral

Commercial systems are designed to achieve complete isolation from the power grid when it faults. Consumer systems tend to lack the relevant circuits (ostensibly for cost reduction reasons) and rely on GFI, breakers, fuses, and/or other safety devices for protection. In the event that those protective devices are activated, the "UPS" will no longer function, but no fire or shock will occur. 

Commercial systems are designed to go further and continue to operate in the presence of grid fault and to contain the grid fault to the external side of the power supply.

In conclusion...

If you have an SPS, the internal connection is already present inside the device and shorting it externally could interfere with safety devices.

If you have a UPS, there are intentional switches to isolate the neutral and bypassing them will disable this protection.

It's not going to help you in the best case; it will hurt you in the worst.