That is studied by Renkin, Montialoux, and Siegenthaler in a recent paper, which is also a job market paper for Tobias Renkin from the University of Zurich. Here is the abstract:
We study the impact of increases in local minimum wages on the dynamics of prices in local grocery stores in the US during the 2001-2012 period. We find a significant impact of increasing minimum wages on prices in grocery stores. Our baseline estimate of the minimum wage elasticity of grocery prices is 0.02. This magnitude is consistent with a full pass-through of cost increases into prices. We show that price adjustments occur mostly in the months following the passage of minimum wage legislation rather than at the actual implementation of higher minimum wages. This forward-looking pattern of price adjustments is qualitatively consistent with pricing models that feature nominal rigidities. We find no differential price effect for products consumed by poorer and richer households, and no evidence for demand effects. Our results suggest that consumers rather than firms bear the cost of minimum wage increases. Moreover, poor households are most negatively affected by the price response. Price increases in grocery stores alone offset at least 10% of the nominal income gains of the poorest households.
Of course this also would suggest the sector is relatively competitive. And if you are wondering, here is the full slate of job candidates from Zurich.