How soon is the responsive layout coming out?

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Are there any log files that I can go through to see the updates and fixes in the plugin?

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There will be some more changes in current AnsPress layout, so after its finalized then I will add responsive style.
for now you can check it here:
https://wordpress.org/plugins/anspress-question-answer/changelog/
I am working to add dedicated roadmap and changelog page in our site.