Solve the generating-function equation for Class 214

Solve the functional equation for the generating function of inversion sequences avoiding the relation triple $(-,\geq,\geq)$, equivalently the patterns $(000,010,100,110,120,210)$, in order to determine the corresponding counting generating function.

Background

For Class 214, the authors describe a generating-tree construction based on labels recording the lengths of two possible consecutive runs of zeros. They derive a functional equation for the associated bivariate generating function $B(z,x,y)$.

Although the equation is relatively compact, the authors do not solve it. The exact enumeration and the algebraic or D-finite status of the generating function therefore remain open; numerical asymptotics suggest growth of algebraic type but do not settle the generating-function problem.