Agents exhibiting generalized control are capable of solving a theme of related tasks, rather than a specific instance. Here, generalized control pertains to the locomotive capacity of quadrupedal animats, evaluated when climbing over walls of varying height to reach a target. In prior work, we showed that Lexicase selection is more effective than other evolutionary algorithms for this wall crossing task. Generalized controllers capable of crossing the majority of wall heights are discovered, even though Lexicase selection does not sample all possible environments per generation. In this work, we further constrain environmental sampling during evolution, examining the resilience of Lexicase to the impoverished conditions. Through restricting the range of samples at given points in time as well as fixing environmental exposure over fractions of evolutionary time, we attempt to increase the ‘adjacency’ of environmental samples, and report on the response of the Lexicase algorithm to the pressure of this reduced environmental diversity. Results indicate that Lexicase is robust, producing viable agents even in considerably challenging conditions. We also see a positive correlation between the number of tiebreak events that occur and the success of individuals in a population, except in the most limiting conditions. We argue that the increased number of tiebreaks is a response to local maxima, and the increased diversity resulting from random selection at this point, is a key driver of the resilience of the Lexicase algorithm. We also show that in extreme cases, this relationship breaks down. We conclude that tiebreaking is an important control mechanism in Lexicase operation, and that the breakdown in performance observed in extreme conditions indicates an inability of the tiebreak mechanism to function effectively where population diversity is unable to reflect environmental diversity.