Can I expect lip corner Botox to correct asymmetry?

When I smile, the left corner of my mouth lifts up well, while the right corner lifts less compared to the left. Since lifting only one side makes me look like I'm sneering, I consciously put more force into the right corner of my mouth when taking photos or smiling. Is it possible to correct this type of mouth corner with Botox?

I heard that getting Botox in the corners of the mouth has the effect of lifting them by weakening the muscles that pull them down. I would like to know if it is possible to get Botox while also correcting any asymmetry.

I am also curious about the treatment cycle and duration of lip corner Botox when the goal is asymmetry correction.

And you know that a characteristic of Botox is that other muscles can develop instead of the injected ones. If I receive treatments long-term, would it help develop the muscles that lift the corners of the mouth upwards, rather than the muscles that pull downwards?