Expected daily range (SPX) and daily realized range

I'm new in quant math, I'm self-studying it. I have two question in exp. daily range topic.

1. How can we make the possibly most accurate estimation for expected daily ranges?

My idea was to take data from yahoo finance, calculate realized vol using garman-klass-yang-zhang formula, then use a model (dunno which one) to calculate an expected trailing seven days avg historical vol for SPX. After that just get the ATM IV and use them both to get daily range. I'm using excel. Is there a way to do that?

1. I've read somewhere months ago a research where a quant scrutinized the VIX performance comparing the VIX projected daily ranges (using $$\text{VIX opening lvl} \times \sqrt\frac{1}{252} \cdot \sqrt\frac{2}{pi}$$ formula) then calculated the daily realized ranges (I don't know the formula, that's what I wanna get). The data was:

(SPX on 3th of January 2022) open $$4778.14$$; high $$4796.64$$; low $$4796.17$$ and close $$4796.56$$ (previous close was at $$4766.19$$), and the research paper got 0.00844149 as daily realized return.

The next day OHLC data was $$4804.51$$, $$4818.62$$, $$4774.27$$, $$4793.53$$, and they got $$0.009251912$$ as daily realized return, 4th January 2022)

Trying to get how the research came up with that numbers I tried different methods: Parkinson took me to 0.004836038; Garman-Klass to $$0.006175798$$ and RS to $$0.005806917$$ if I put it in well. Yang Zhang also took me to $$0.005742242$$ however I'm not sure if I put in the formula correctly.

The closest numbers I got by $$\ln\left(\frac{\text{high}}{\text{low}}\right)$$ but not exactly the same. With VIX I got the same values the paper said using the formula.

Anyone have any clue what the calculation could have been? I'm not sure in mine, are those numbers correct if you check the data?