It cannot be true "no matter what annuity or growth rates are". If it's true in practice, it must be because there is a sensible relationship between growth rate and annuity rate, and between annuity rates year-on-year.
Suppose growth rate is 6% and annuity rate this year 7% and next year it's 7.5%. Suppose your AVC fund is £20k.
Then option 1 will buy an annuity paying £1400pa, whilst a 1 year delay will first grow the AVC fund to £21200 and will then buy an annuity paying £1590pa. But to buy an annuity paying £190pa next year would cost £2533. You'd need to be an exceptionally astute investor to turn £1400 into £2533 in just one year, an 81% growth, especially if you only get the £1400 in monthly chunks.
This probably just means that the 6%/7%/7.5% triple is unrealistic.