Is there any scientific study that shows that T improves with a ketogenic diet?
I couldn't find any...
Here is evidence that it might improve with healthier diet
http://www.tandfonline.com/doi/abs/10.1080/14992027.2017.1331049
Here is how they defined "healthy diet" in this study:
"For each of the five food group components (meat,
dairy, fruits, vegetables and grains), individuals who consumed the
recommended number of servings received a score of 10; scores
decreased for those with less than the recommended intake; greater
than recommended intake of the five food groups (meat, dairy,
fruits, vegetables and grains), did not reduce the category score. Fat,
saturated fat, cholesterol, and sodium were scored based on
consumption of no more than the recommended maximum intake.
If fat and saturated fat intake were 30% of total caloric intake or less
(at or below the maximum recommended intake), a score of 10 was
received for each; if cholesterol and sodium intake were at or below
maximum recommended intake values (300 mg cholesterol,
2400 mg sodium), a score of 10 was received for each. Higher
than the recommended intake of fat, saturated fat, cholesterol, or
sodium decreased scores. Finally, for dietary variety, a score of 10
was given if at least 16 different foods were consumed, decreasing
to zero if 6 or less different foods were consumed."