Implications of a taxable Traditional-to-Roth conversion and a 'backdoor' Roth conversion in the same year.

Here is the scenario:

  1. I've had a Traditional IRA with Vanguard since 2016 that started out as a Rollover from an employer-sponsored retirement plan. I've been making regular contributions to that throughout the years such that it grew to ~$40K and contains some combination of deductible and non-deductible contributions  + earnings. This total includes ~$1.5K in non-deductible contributions for 2022 (meaning I still have $4500 I can contribute towards the $6K limit)
  2. In July 2022 I decided to convert the entire balance to a Roth IRA knowing that I will have to pay taxes on that when I file next year. The impetus for this move was to gain the ability to perform backdoor conversions in the future since every year the initial balance of the Traditional IRA will be $0
  3. I would now like to open a new Traditional IRA this year (2022), fund it with the remaining $4500 using post-tax money, then convert all of it to the Roth IRA, i.e. the backdoor conversion.

My question is this: will the two conversions described in steps 2 and 3 be combined in the eyes of the IRS such that the pro-rated tax applicable to step 2 will also apply to the $4500 in step 3? Or does that somehow get handled differently at tax time from the first conversion (step 2) such that the $4500 in step 3 is not subject to any tax?

 

Thank you!