Link to paper: https://papers.ssrn.com/sol3/papers.cfm ... id=2658343

Getting confused at one point in this paper:
Context: 
Section, IV, Equation 4 shows a regression form of the minimum variance delta hedge. On page 9 (at the top), it is mentioned that the resulting delta_mv - delta_bs is approximately quadratic in delta. I'm able to derive the result and graph. I understand that this is an attempt to estimate a reasonable functional form for the delta_mv - delta_bs
In equation 5 however, it is shown that the delta_mv - delta_bs = scale_factor * quadratic regression function. The scale factor is a scaled vega. I'm able to follow the derivation of this as well (from the appendix etc.; independent of equation 4)
Question: I'm not able to reconcile equation 4 and 5:

1. Eq4: estimating the delta_mv from the regression and subtracting delta_bs gives me ~quadratic function; mentioned on page 9 (top) and confirmed. 
2. But from equation 5, the same LHS term, (delta_mv - delta_bs) involves a scaling factor AND the quadratic function. This is not consistent with the regression equation 4.
3. From equation 5, I would guess that the expression (delta_mv - delta_bs)/scale factor ~= quadratic equation. That would make the equation 4 and 5 connect directly, but that's not the case; HW state that the un-scaled diff; delta_mv - delta_bs is ~ quadratic; not the scaled difference.

Would appreciate some advice from the seniors!

Thanks!
PS - I'm aware that this paper has been part of the final projects in some cohorts. I already finished the CQF in 2022; this is purely out of my own interest.

Continuing with the paper - I was able to replicate the delta_mv - delta_bs. This appears to be reasonably quadratic. Then, following expression 5, I also tried the scaled expression: (delta_mv - delta_bs) * S*sqrt(T)/Vega_BS; I'm expecting this to be quadratic too....but it looks nothing like a quadratic function (the graph at the bottom). Does anyone else have any experience with this topic?