With Quark version 3.4-406 the game continuously runs in the background but wont actually open. No crashlogs or files are produced.

I have the same issue, I found the cause of it in the debug.log:
[15Aug2023 12:49:32.696] [main/FATAL] [mixin/]: Mixin apply failed quark.mixins.json:client.ItemRendererMixin -> net.minecraft.client.renderer.entity.ItemRenderer: org.spongepowered.asm.mixin.injection.throwables.InvalidInjectionException @at("CONSTANT") on net/minecraft/client/renderer/entity/ItemRenderer::renderQuads with priority 1000 cannot inject into net/minecraft/client/renderer/entity/ItemRenderer::m_115162

debug.log

For me with only Forge 43.2.21 Quark 3.4-407 ARL 1.8.2-55, Debug.log ends at:
[15Aug2023 08:57:16.027] [pool-3-thread-1/WARN] [mixin/]: Error loading class: com/llamalad7/mixinextras/sugar/ref/LocalRef (java.lang.ClassNotFoundException: com.llamalad7.mixinextras.sugar.ref.LocalRef)
Not sure if this is a different issue

Same here, forge 43.2.14, quark 3.4-407

EDIT: For me, debug.log seems to be hanging at:
[15Aug2023 07:41:42.797] [main/DEBUG] [mixin/]: Mixing client.ItemRendererMixin from quark.mixins.json into net.minecraft.client.renderer.entity.ItemRenderer so the above seems to apply to me too

8f66fb3 should fix this

Issue persists with Quark-3.4-408