To bring Ben and Aki’s comments into context – convergence diagnostics like \hat{R} consider the marginal behavior of your chains, and that can be very different for different variables. Some variables converge quickly, and their expectation values can be estimated quickly, while some converge more slowly and require longer running times.
If \hat{R} is close enough to one for all of your variables except for lp__ then the expectation value estimates for those variables are probably okay, especially if none of the other diagnostics are indicating problems.
But it doesn’t mean that the estimates for expectations of any function of those variables will be okay! Variables can correlate with each other in ways that makes the convergence of a function worse and hence the estimate untrustworthy.
lp__ tends to be extremely sensitive to the autocorrelation of the Markov chain and hence provides a reasonable bound on how well any function of the variables will converge. In other words, ensuring that the diagnostics for lp__ are good gives you the strongest evidence that your fit is okay but if you focus only on a few variables and carefully check the diagnostics for those variables then you may be able to ignore lp__ for that very specific context.