I understand that many years have passed, and this paper can't be shared, but since you read it, and you've been so kind to summarize it here, can I ask you a couple of questions below?
Then express [$] dC [$] in terms of Black-Scholes Greeks. Once you've done that you have an equation for implied volatility. If you then assume that the correlation between implied volatility and the stock/index is constant across strikes, and also that the dollar gamma, volga and vanna are constant across strikes you have basically a way to calibrate the smile using 4 pillar options (or 3 if you assume the drift of implied vol is zero). This is in essence the so-called GVV cost frameworkby Arslan et al on which the paper by Alexander Giryavets builds.
With "the correlation between implied volatility and the stock/index is constant across strikes", do you mean the fixed strike BS volatility v, i.e. E[dv ds] constant across strikes or E[dv/v * ds/s] constant across strikes like in the Arslan paper?
You also say that "that the dollar gamma, volga and vanna are constant across strikes". How can it be? Dollar gamma is maximum at the money, it cannot be constant across strikes.
Last question: does the paper deal with skew and fixed-strike floating-strike volatility?
Thank you in advance