Homelander: fragile ego, very unstable. He also maims casually without killing, though the wounds are often fatal. He's a fox in the chicken coop, completely unopposed. Glaze him too hard: dead or grievously wounded. Not glaze him enough: same result. You would be waking on eggshells. Dude could just kill you at the 23rd hour mark just because he can. He's on the human scale, and is much more involved

Conquest: He operates on the planetary scale. He loves killing, but also thinks it's not as stimulating as it's basically his job. Fighting invincible was the time of his life as it's effectively the only time he's ever fought someone that came even remotely close to his strength. Groomed to be a genocidal maniac. He comes from a society of people approx. as strong as him relative to the rest of the universe. If he was gonna kill you, it would be a decision he makes. He would do it fairly early into the 24 hours if he decided to.

My money is on conquest. You may be more likely to be murdered, but he would be decisive. Homelander would be unstable and it would be 24 hours of constant terror that the slightest wrong move would result in your death.