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This is a great tax planning question. Your AGI will be increased by the amount of the conversion; it will not stay the same as if no conversion occurred. Whether your conversion will be taxed at the same rate as your AGI prior to the conversion depends on where your taxable income falls with the IRS Tax Brackets. Please note that your AGI is your total income minus adjustments while your Taxable Income is your AGI minus Deductions.  Your Tax is figured using your Taxable Income. 

The example I am providing will have 2 scenarios using the 2022 Tax Bracket for a single taxpayer: 

Scenario 1 - Conversion Does Not Change Tax Bracket

In this scenario, assume a single taxpayer with taxable income of $25,000 and a $5,000 Roth conversion for tax year 2022. 

Before Conversion: Taxable Income is $25,000; Tax Bracket = 12%

After Conversion: Taxable Income is $30,000; Tax Bracket = 12%

Scenario2 - Conversion Does  Change Tax Bracket

In this scenario, assume a single taxpayer with taxable income of $25,000 and a $5,000 Roth conversion for tax year 2022. 

Before Conversion: Taxable Income is $40,000; Tax Bracket = 12%

After Conversion: Taxable Income is $45,000; Tax Bracket = 22%

Note: The US has a progressive tax rate. In the latter example, only the amount over $41,775 will be taxed at the 22% rate.