The six operations $f_!,f^!,f_*,f^*,\otimes,\mathcal Hom$ have the property that they preserve estimates on weights in one direction.

For $f_!,f^!,f_*,f^*$ I can see, that they don't preserve purity general. One can also give sufficient criteria for them to preserve purity, by ensuring them to be "equivalent" to their dual (smooth pullback, proper pushforward...).

What about the other operations $\otimes,\mathcal Hom$? What is an example of two pure sheaves, such that say $\cal F \otimes \cal G$ is not pure anymore?

Are there condtions that force $\otimes,\mathcal Hom$ to preserve purity? (By making them equivalent to their dual?)

I am most interested in the following situation:

Let $IC_x$ and $IC_y$ be two Bruhat constructible $IC$-sheaves on a partial flag variety. Then it is well known, that the space $Ext^{\bullet}(IC_x,IC_y)$ is pure. Does the sheaf version of this statement hold as well? Is the sheaf ${\cal Hom}(IC_x,IC_y)$ pure?