Which 4K monitor for 2013 Mac Pro???

Mac Pro (Late 2013) - Technical Specifications - Apple Support

Your Mac has six Thunderbolt 2 ports. These are old-style Thunderbolt ports with Mini DisplayPort connectors that can double as Mini DisplayPorts. It also has a HDMI version 1.4 port.

Although your Mac can drive three 5K displays, they must be the type that accept input on two physical cables. As far as I know, there are no adapters that you can use to combine two Thunderbolt 2 outputs from the 2013 Mac Pro into a single Thunderbolt 3 output for a modern 27" 5120x2880 pixel display (Apple, LG, or Samsung).

Neither Apple's Technical Specifications nor MacTracker indicate the maximum resolution for a connection involving a single Thunderbolt 2 port (whether operating as a Thunderbolt 2 port, or as a Mini DisplayPort).

MacTracker claims that the maximum resolution on the HDMI port is "up to 4096 by 2160 pixels at 24 Hz."

A Google search led to this Apple Discussions thread

2013 Mac Pro Maximum Resolution - Apple Community

It has a post by Mr. Bennet-Alder which links to the archived Apple Support article

Use multiple displays with your Mac Pro (Late 2013) - Apple Support

which describes the "Display configurations you can use with your Mac Pro (Late 2013)." This article does not say anything about the possibility of using two Thunderbolt 2 outputs to support one dual-cable 5K display – perhaps since all of the 5K displays that accepted that form of input were discontinued years ago.

Tthe bandwidth available over a ThunderBolt-2 Bus is limited to nominal 20 G bits/sec.

The bandwidth required to support 4K at 60 or 75 Hz is 17.25 G bits/sec (HBR2), and Apple says that 4K should be supported over ThunderBolt.

...

That one should work fine. Keep your cablers under 1 meter, and don't believe the shipped in the box cable if it gives you any trouble.

Consumer Reports liked ASUS PA28QV as a good HD monitor. its big brother is available for US$388.63, B&H is closing at 2 PM today Tues 2 Oct for Rash Hashana

https://www.bhphotovideo.com/c/product/1763798-REG/asus_proart_display_pa279cv_27.html

DisplayPort Displays at 4K up to 75Hz are running at 17.28 Gbits/sec. 2 meters is the absolute upper limit.

Anything higher requires a bump up in speed to 25.92 Gbits/sec, which definitely requires cables under a meter. It turns out DisplayPort over ThunderBolt connection instead of DisplayPort does not improve the cable lengths, which are limited to 0.8 meters for super-engineered Apple brand cables, and only 0.5 meters for brand X Thunderbolt cables.

HDMI uses 5 Volt signal levels, up to and including 4K displays at 60 Hz. HDMI uses a signing method unique to HDMI and Single-Link DVI.

Above 4K at 60 Hz, HDMI switches to the same signaling method used by DisplayPort, but requires ULTRA cables. There is no specification on the maximum cable length, but if it works over about a meter of cable I would be VERY surprised, because nothing else using that method at that data rate works on over a meter of cable..

<< If I use the Apple Thunderbolt 3 to Thunderbolt 2 adapter, will I be able to use this monitor in 4K? >>

You could drive up to three displays like that at 5K, provided your displays are genuine ThunderBolt. The Apple ThunderBolt-3 <-> ThunderBolt-2 adapter passes Thunderbolt, ONLY. In addition to the Thunderbolt adapter, I think you may need a ThunderBolt-2 cable, and they can be pricey as well.

Originally, this model reportedly supported as many as three 4K displays or six Thunderbolt displays. However, on June 16, 2015, without updating the graphics cards themselves, Apple quietly increased the official support to as many as three 5K displays (5120x2880) -- two using Thunderbolt 2 ports and one using the HDMI port. The support for as many as six 2560x1600 displays using all six Thunderbolt 2 ports remained unchanged.

from:

https://everymac.com/systems/apple/mac_pro/specs/mac-pro-quad-core-3.7-xeon-e5-gray-black-cylinder-late-2013-specs.html

<<Although your Mac can drive three 5K displays, they must be the type that accept input on two physical cables. >>

that's a very subtle, but very important distinction!

so for 5K, that limits you to displays with TWO DisplayPort inputs and Picture-By-Picture feature in the display, as the only way to get to 5K.