Multi venue strategy: okay to use historical data from just one venue during back testing?

I'm looking to trade spot FX across four different venues. I have conducted historical analysis on tick data from just one venue only (LMAX Global). If I want to use the exact same strategy across the four different venues, do I need to re-do back testing with tick data from each individual venue? Or can I make an assumption that the historical price data for the chosen FX pair will be relatively similar across all four venues? Only best bid and ask tick data will be analysed (historical volume data will be ignored).