  We discuss our conjecture for simply laced Lie algebras level two string
functions of mark one fundamental weights and prove it for the $SO(2r)$
algebra. To prove our conjecture we introduce $q$-diagrams and examine the
diagrammatic interpretations of known identities by Euler, Cauchy, Heine,
Jacobi and Ramanujan. Interestingly, the diagrammatic approach implies these
identities are related in the sense that they represent the first few terms in
an infinite series of diagrammatic identities. Furthermore, these diagrammatic
identities entail all the identities needed to prove our conjecture as well as
generalise it to all $SO(2r)$ level two string functions. As such, our main
objective is proving these series of diagrammatic identities thus extending the
works mentioned and establishing our conjecture for the $SO(2r)$ level two
string functions.
