TLDR: The winner is the Wither for reasons discussed below. 1.19 came out 2 days ago and this question was just itching to be settled. I do want to get this out of the way that the Java Wither is a pushover, hardly worth even talking about. I ran two main tests, one for Normal Mode and one for Hard Mode. (The normal mode test was an accident but it ended up being worth it.) 4 battles for normal mode and 2 for hard took place. I had the wither spawn in an area with an already active Warden. I felt like this would be the more realistic setup. I tried the fight in a mesa sand area, mesa highlands, and acacia plains. The beginning of the fight always began in the Wither's favor. Wither would get several shots off against the Warden before it got angry enough to return fire. Once angry enough to attack it would take an additional couple of seconds to begin firing off sonic booms. Sonic booms took a small chunk out of the Wither's health and were unavoidable, however there was a delay between shots. In a location unsuitable for the Warden like the Mesa sands, falling sand blocks actually delayed or even canceled the sonic booms as well as forced the Warden to readjust his position while still getting rained on by skulls. In the acacia plains, the Wither actually got stuck in the occasional tree, allowing the Warden to get a punch or two in on the Wither. The mesa highlands were the Warden's favorite because of the blast resistance of the floor. As long as the Warden had somewhat stable footing, it could continue shooting at the wither. The turning point was reached once the Wither dipped to the halfway point. I was curious about how the wither skeletons would affect the fight. Would they start attacking the Warden? No they actually did nothing. The Wither however would change its strategy and go down low for the kill. This turned out to be a mistake. The Warden who could now punch the Warden would rush straight in and start pounding the thing to death. The blows being stronger than the sonic booms and lacking the delay finished the fight quickly. The Wither's dash attack would disorient the Warden for a few seconds but the speed and ferocity of the blows proved to be the Wither's undoing. In fact this proved to be a constant. All four of the fights went to the Warden on Normal. One fight left the Warden with as much as 13 unenchanted netherite sword swings worth of health left, wow. Hard Mode proved to be a different story. Both contenders would deal more damage so wouldn't the Warden push the Wither into the halfway point faster and finish it quickly? Well actually no. I actually saw little/no difference in sonic boom damage. So while the Warden's DPS failed to accelerate, the Wither's absolutely did. This turned out to be fatal, the Wither's hyperfast skulls would silently obliterate the Warden's health. Though in both fights, the Warden could still get the Wither below half, the damage output of the Wither overcame. And with a 1/4 of max health too. As Hard mode is essential to villager mechanics working correctly, I feel like the Wither won overall. In conclusion, if you want your warden guard dog to stay alive against the Wither, stick to Normal or Easy, or play Java edition.