I've found a published journal titled "Educational Leadership" that thoroughly explores with supportive data of the arguments for and against homework, available here: http://www.ascd.org/publications/educational-leadership/mar07/vol64/num06/The-Case-For-and-Against-Homework.aspx
Based on the above reference, academic improvement with assigned homework has technically been proven. It also shows age does impact the margin of improvement. My assessment of the data is that the older the child, the greater the margin of improvement. The data contained in the above reference also states that "... research has produced no clear-cut consensus on the benefits of homework at the early elementary grade levels". Note that the youngest students data is cited for is for grades 4-6, at a meager performance gain of 6%.
Additionally, I also found this article written by Harris Cooper, with relevant subject matter expertise. Harris Cooper is, according to the Editor's Note:
"Harris Cooper is professor of psychology and neuroscience at Duke,
where he also directs the university's Program in Education, and is
author of "The Battle over Homework: Common Ground for Administrators,
Teachers, and Parents" (Corwin Press)
The article available here: https://today.duke.edu/2006/09/homework_oped.html
Cooper expands on how age plays a part in academic improvement with age when he writes:
The homework question is best answered by comparing students who are
assigned homework with students assigned no homework but who are
similar in other ways. The results of such studies suggest that
homework can improve students' scores on the class tests that come at
the end of a topic. Students assigned homework in 2nd grade did better
on math, 3rd and 4th graders did better on English skills and
vocabulary, 5th graders on social studies, 9th through 12th graders on
American history, and 12th graders on Shakespeare.
Based on the above references, the teacher's statement: "Research has been unable to prove that homework improves student performance" is technically inaccurate. It would be more accurate to state "research shows homework assigned to early elementary students has negligible academic improvement".
More context is honestly needed, such as what subject(s) matter the teacher is responsible for. ( I had two teachers in 2nd grade, subjects were split between them.) As Cooper found research showing measurable improvement in mathematics among second grade students, the teacher's decision to not assign homework is only supported by research as long as she doesn't teach math.