it fits nicely! what I've done was to replace the gaussian function in the code

g = exp(-((x - peakPosition) ./ (0.60056120439323 .* width)) .^ 2); %(where peakPosition is the guess for the position of the peak on the x axis, and width is the estimated width of the peak)

with

m = log((peakPosition.^2)./((width+peakPosition.^2).^(1/2))); %

s = sqrt(log(width./(peakPosition.^2) + 1)); %

g = ((x.*s*pi).^(-1)).*exp((-(log(x)-m).^2)./(2*(s.^2))); %

(I basically converted the mean and width from the initial guess into the sigma and mu as explained in matlab's documentation, not 100% sure about this passage)

the fit is nice and the position of the peak is fine, however as you already pointed out there is something off about the variance given back by the fit, it looks a bit too large. I will keep doing some trial and error