Neither an acute increase in testosterone after strength training, nor a long-term basal increase, appears to be necessary to experience gains in strength and hypertrophy.
However, the increase in testosterone that occurs just after strength training is followed by an increase in the muscle androgen receptor, while there is a subsequent drop to basal levels in testosterone (by binding to its receptor to exert its action).
This combination of responses (increased muscle testosterone receptor and low blood concentration of testosterone) may be indicative of a shift of testosterone from blood to muscle after strength training, which may play a key role in optimizing strength and hypertrophy.
When we compare testosterone and androgen receptor after strength training in the fasted or fed state, we see that in the fasted state (graph a), there is an increase in testosterone followed by a return to baseline after strength training, but no change in androgen receptor 60 minutes after training. However, when training in a post-prandial state, i.e., having eaten (Graph b), there is a further increase in testosterone just after training, but shortly thereafter it drops dramatically. This reduction in circulating testosterone in the blood correlates with an increase in androgen receptor content (Figure b), which indicates that there is a shift of testosterone into the muscle, where it will bind to the androgen receptor favoring certain pathways that will lead to muscle optimization in terms of strength and hypertrophy.
To make it simpler: according to this study, training on an empty stomach produces a lower muscle anabolic state derived from the testosterone-receptor interaction compared to training having eaten beforehand. However, further research is needed to draw firm conclusions.