Is there co-movement of agricultural commodities futures prices and crude oil?

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Forecasting realized volatility of agricultural commodity futures with infinite Hidden Markov HAR models (2019, background): “This is also found for an earlier period in Natanelov et al. (2011) in a co-movement relationship of agricultural futures prices and crude oil.”
Biofuel-related price transmission literature: A review (2013, methodology): “A rather puzzling result in Natalenov et al. (2011) is that long-run causality is generally found to flow from food commodities towards crude oil.”
Market interdependence and volatility transmission among major crops (2016, methodology): “We include variables widely used in previous studies on price co-movement (for example, Deb, Trivedi, and Varangis 1996; Ai, Chatrath, and Song 2006; Le Pen and Sévi 2010; Gilbert 2010; Natanelov et al. 2011), conditional also on being available on a daily, weekly, and monthly basis.”