Do plastid-related characters support the chromalveolate hypothesis?

According to the idea of secondary endosymbiosis, plastids with three and four envelope membranes have evolved from either red or green algal endosymbionts engulfed by phagotrophic protozoans. Although this hypothesis is nowadays commonly accepted, the number of secondary endosymbioses still remains controversial. One of the models, known as the "chromalveolate" hypothesis, postulates that the 4 membrane-bound plastids of Chromista and the 3 or 4 membrane-bound plastids of Alveolata result from a single secondary endosymbiosis involving a rhodophyte as the endosymbiont. Although this model has found many followers, a variety of data clearly contradict it. The ideas that became the direct inspiration for formulation of the "chromalveolate" hypothesis are also now questionable. In this comment, I discuss all these problems in the light of the most recent phylogenetic, cytological, and genomic data.