To begin with, the noun rank has nothing to do with the adjective rank.
The noun is of French origin (possibly Frankish ultimately) and originally denoted a series or line of things (range has the same origin); then specifically a line of warriors or soldiers (whence our 'rank and file'); then the successive lines, the first, second, third ranks, and so forth, in which soldiers are deployed; and finally, by way of one's position in the front, middle or rear rank, one's grade or standing.
The adjective rank is of Germanic origin; the precise line of descent is obscure but it seems originally to have denoted various attributes of the noble warrior: proud, haughty, rebellious, strong, vigorous, full-grown. In ME the martial connection faded and the word became used particularly of undesired and excessive strength, vigor and abundance; it was extended to vegetation (we still speak of 'rank weeds') and to loathsome behavior and smells (O mine offense is rank says Claudius); eventually it acquired such a strong negative connotation that came to mean 'corrupt, foul, festering', and eventually simply 'extremely (offensive)'.
See the Oxford English Dictionary, 1st edition.
When your source writes of rank dishonesty it probably does not intend any one of these later senses but (at some level) all of them simultaneously: extremely gross, corrupt, loathsome dishonesty.
This is close to being a dictionary question. Wiktionary gives this relevant definition for "grave":
Serious, in a negative sense; important, formidable. [from 19th c.]
This pretty much covers it. However, what the dictionary doesn't say is that this is a "sticky" adjective. (I'm sure there's a more technical grammar term for it, but I don't know it).
What that means is that you will almost always hear "grave" used to modify a small number of words, notably "injustice" or "insult" or "error." They are not quite set phrases or cliches, but they have "stuck" together.
So while you will never hear a native speaker say:
I had a grave realization this morning.
and you would be unlikely to hear even something like:
Gather round; I'm afraid I have grave news.
you might well hear someone say:
The lack of celery in your beef jerky is a grave injustice to vegetarians.
Best Answer
It's a phrase from journalism, meaning withdraw the story from publication, as when a story in an early edition of a newspaper proves to be inaccurate or unfounded (or annoys a big advertiser) and is removed from subsequent editions.
Over the years the phrase has been extended to broadcast and internet journalism.