I have discrete optical spectra data consisting of pairs of wavelengths and intensities, and I would like to make a ListPlot in which the points are both joined, and colored according to the wavelength as is the filling.

Here is an example.

Here's fake data that illustrates the problem:

ListPlot[ Table[Style[{λ, Sin[λ/400]}, ColorData["VisibleSpectrum"][λ]], {λ, 400, 700, 5}], PlotStyle -> PointSize[Medium], Filling -> Axis, FillingStyle -> Function[{λ, y}, ColorData["VisibleSpectrum"][#[[1]] &]]] 

I can get the points to be properly colored, but not the filling. Moreover, the ideal is to have a continuous filling underneath curve (as in the linked example), i.e., not discrete bars. I've tried numerous variations on the FillingStyle, without success.