We use a superb five-hour sequence of 900 solar images taken at La Palma to study long-duration persistence in the solar granulation, in the context of the long-lived "intergranular holes" discovered by Roudier et al. (1997) and the contention that these mark sites of convective downflow plumes. We develop a procedure to locate "persistency regions" that contain granular brightness maxima or minima over extended periods (up to 45 min), while allowing for lateral drifts due to horizontal flows. Statistical evaluation of the co-location probability for different pixel classes is first used to quantify the likelihood of long-term stationarity for different granular brightness classes and for the persistency regions, and then to evaluate the amount of preferential alignment, at different frequencies and time delays, between excessive Fourier modulation and granular brightness and persistence. The results support the existence of long-lived intergranular holes. There is large persistency difference between the brightest and the darkest features; some of the latter have location memories as long as two hours. In addition, the darkest intergranular features are found to be sites of enhanced Fourier modulation in the 3-min acoustic regime, improving earlier results through much higher statistical significance. However, the persistency regions containing intergranular holes do not seem to produce the excess acoustic emission that would be expected above downflow plumes.