Wouldn’t Underwood’s plan for Sharp in season 3 of House of Cards be obvious?

In episode 11 of season 3 of House of Cards, would it not have been obvious that Sharp was “in concert” with Underwood if she had announced her support for him, as planned, rather than for Dunbar, after dropping out? It’s explicitly pointed out that Underwood wants to avoid such an impression:

Underwood: Oh you mean the thing about your kids?
Sharp: That wasn’t what we discussed.
Underwood: Well I had to hit you hard so she wouldn’t. And besides, we had to dispel any suspicion that you and I might be in concert.

But wouldn’t that still have been obvious if she had announced her support for him, especially if Underwood had later, as planned, picked her up as his vice-presidential candidate?

During the first Democratic nomination debate, Sharp is highly critical of Dunbar, and somewhere between non-critical and outright supportive of Underwood. Underwood does hit her with the above-mentioned “thing”. But other than that, and while we of course don’t see the whole debate, I still did not get the impression that there was much in the way of cross-fire between them. In fact, when Underwood makes the comment about her stepchildren, she does not just seem struck that he’s using that point against her, she seems utterly flabbergasted he’s being critical of her at all. As if being critical of each other was entirely not part of the plan. If Sharp truly wanted to be president, as the public is supposed to believe, I would expect her to attack both opposing candidates, perhaps even focussing more on Underwood than on Dunbar with him being the sitting president.

I’m not that familiar with American politics, but it seems like a strange sequence of events to me for one candidate to focus criticism on the one other candidate that is not the sitting president, then to drop out after only the first debate and get picked up as the vice-presidential candidate later. Wouldn’t that make the first debate a rather obviously “staged” unfair fight? Or perhaps the answer is that it would be obvious but by the time it is so, the public wouldn’t care much?