NC Poincaré problem: equality of M_d and M_d^C

Prove that the universal skew field M_d of O_d coincides with M_d^C, the universal skew field of fractions of the ring O_d^C of NC functions admitting compact realizations; equivalently, show that every uniformly meromorphic NC function with a compact realization is representable as a non-commutative rational expression in NC entire functions.

Background

The authors introduce a second skew field M_d^C arising from compact operator realizations and note that O_d embeds into O_d^C. In one variable, M_1=M_1^C; however, for d>1 this equality remains unclear.

Establishing M_d=M_d^C would provide a non-commutative analogue of Poincaré’s classical problem that every meromorphic function on C^d is a quotient of entire functions, adapted to the NC meromorphic and entire frameworks developed in the paper.