Density-lines in PCA plot? Short version:
Someone wants me to draw "density-lines" onto a PCA plot for every point on the PCA plot. Which does not make sense in my understanding.
PCA is transforming the high dimensional data to principal components. How am I supposed to add a "density" for every point on a plot showing the two most relevant principal components?
Longer version:

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*I have high-dimensional data. Gene expression data, to be precise (about n = 22000 dimensions). The data is from k biological samples, so every sample has values for the about 22'000 dimensions.

*Now I calculate the principal components and show the two principal components which explain most of the variation. The k samples now fall on the 2D plane reflecting the two most relevant principal components as they are calculated for the respective sample.

*My collaborator now wants "denisty-curves" on the PCA plot for every one of the k points. Which does not work for a PCA plot, right? We reduce the n-dimensional space to fewer dimensions and then project the data onto this (in my case) two principal components. So in my understanding density lines for the reduced data points do not make sense, do you agree?

Example -- Let's assume a PCA plot as in the first figure here: https://cran.r-project.org/web/packages/ggfortify/vignettes/plot_pca.html
Would you say that some sort of density lines for every point reflecting the underlying data does make sense in this case?

It sounds like there are two problems: one of methodology and another of conflict resolution. My initial impression is say to just "split the baby". At the fundamental level, the collaborator is just asking for a simpler way to visualize the results of the PCA so what could be the harm in delivering a "prettier graph."  At the same time, everyone needs to be well aware as you rightly point out no one should try to make any inferences using the resulting marginals.
To that end, one proposed way to demonstrate the issue to the collaborator is to run a sensitivity analysis on the data to illustrate how much potential there is for estimation error in the method they have requested.  You can bootstrap several samples or add varying degrees of noise and inspect how different the plots are between the full dataset and the augmented ones.
Here is an article on unsupervised dimensionality reduction on gene expression data which found that PCA was relatively robust in this regard: Comparative study of unsupervised dimension
reduction techniques for the visualization of
microarray gene expression data. That said, in Figure 6 of their paper they found that with a very low number of principal components, the issue you described can be very impactful.
