A short proof of a Landen product identity for the Gr\"otzsch ring function

We give a short, self-contained proof of a closed form for the infinite product associated with the Landen transformation of the Gr"otzsch ring function, combining the arithmetic-geometric mean, theta duplication identities, and an elementary coefficient comparison. The identity itself is the equality case of a 1999 product theorem of Qiu and Vuorinen, and we record what is known and what apparently cannot exist about its extensions.