Time for the radical centrist answer haha
While I agree a great design can improve conversions it's really one side (or one color) of the Rubik's cube.
Strong copy, increasing traffic, creating content, marketing, etc also contribute to conversion. You first need steady traffic to even be able to measure conversion rate in a meaningful way.
If you get 100 visitors per month and 1 conversion normally, and then you change the design and you got 2 conversions this month, is that enough data to really point to the design being the differentiating factor? I'd argue no.
Now if you get 10 conversions on average for a few months after changing the design then clearly the design made a huge impact.