In a separable Hilbert space $\mathcalH$, greedy algorithms iteratively define $m$-term approximants to a given vector from a complete redundant dictionary $\Dict$. With very large dictionaries, the pure greedy algorithm cannot be implemented and must be replaced with a weak greedy algorithm. In numerical applications, \em partially greedy algorithms have been introduced to reduce the numerical complexity. A conjecture about their convergence arises naturally from the observation of numerical experiments. We introduce, study and disprove this conjecture.