Design and Defend Priors for a New-Checkout Conversion Test

Your site's checkout conversion has hovered around 3.2 percent for a year. You will test a redesign with 4,000 visitors per arm, expecting 3.75 percent (150 conversions) for control and 4.375 percent (175) for the redesign. A colleague says "just use a flat prior, it is objective". Another says "use a Beta(32, 968) prior, we know the rate".

1. Propose a prior for both arms. Express its strength in pseudo-observations and say how you chose it.
2. Use a prior predictive check to compare the flat prior, your prior and the Beta(32, 968) prior.
3. Show the effect of the three priors on P(B \gt A) and describe the sensitivity analysis you would report.
4. Why should the prior be the same for both arms?