A truncated Jacobi triple product theorem

Recently, G.E. Andrews and M. Merca considered a truncated version of Euler's pentagonal number theorem and obtained a nonnegativity result. They asked the same question on a truncated Jacobi triple product identity, which can be found as a conjecture in a paper of V.J.W. Guo and J. Zeng. In this paper, we provide an answer to the question, which is purely combinatorial. We also provide a combinatorial proof of the main theorem in the paper of Andrews and Merca.